SQU Journal for Science, 2019, 24(1), 57-70  DOI: 10.24200/squjs.vol24iss1pp57-70 

Sultan Qaboos University  

57 

 

Two Groups 𝟐𝟑.𝑷𝑺𝑳𝟐(𝟕) and 𝟐
𝟑: 𝑷𝑺𝑳𝟐(𝟕) of 

Order 1344   

Mehmet Koca
1
, Ramazan Koc

2
 and Nazife O. Koca

3
*

 

1
Retired Professor; 

2
Department of Physics, Faculty of Engineering, Gaziantep University, 27310 

Gaziantep, Turkey; 
3
Department of Physics, College of Science, Sultan Qaboos University, P.O. 

Box 36, PC 123, Al-Khod, Muscat, Sultanate of Oman. *E-mail: nazife@squ.edu.om. 

ABSTRACT: We analyze the group structures of two groups of order 1344 which are respectively non-split and split 

extensions of the elementary Abelian group of order 8 by its automorphism group 𝑃𝑆𝐿2 (7). Two groups have the same 
number of conjugacy classes and the set of dimensions of irreducible representations is equal. The group  23.𝑃𝑆𝐿2(7) 
is a finite subgroup of the Lie Group  𝐺2 preserving the set of octonions ±𝑒𝑖 , (𝑖 = 1,2, … ,7) representing a 7-
dimensional octahedron. Its  three maximal subgroups  23: 7: 3,  23. 𝑆4 and 4. 𝑆4: 2 correspond to the finite subgroups of 
the Lie groups 𝐺2, 𝑆𝑂(4) and 𝑆𝑈(3) respectively. The group 2

3: 𝑃𝑆𝐿2(7) representing the split extension possesses 
five maximal subgroups 23: 7: 3,  23: 𝑆4, 4: 𝑆4: 2 and two non-conjugate Klein’s group 𝑃𝑆𝐿2(7). The character tables 
of the groups and their maximal subgroups, tensor products and decompositions of their irreducible representations 

under the relevant maximal subgroups are identified. Possible implications in physics are discussed. 

 

Keywords: Finite groups; Discrete octonions; Group extensions; Character table; Tensor products. 

𝟐و  𝑷𝑺𝑳𝟐(𝟕).𝟐𝟑مجموعتين
𝟑: 𝑷𝑺𝑳𝟐(𝟕)  4411عدد عناصرها 

محمد كوجا
4

، رمضان كوك
2 

نزيفة أوزيدس كوجا و
4  

وهن على الترتيب االمتداد المقسوم وغير المقسوم لمجموعة األبيليان االبتدائية ذات  4411قمنا بتحليل الشكل لمجموعتين عدد عناصرهن :صلخمال

. .المجموعتين يمتلكن نفس العدد من كوجنسي كالسس وأيضا أبعادهن متساوية في تمثيل الريديوسبل𝑃𝑆𝐿2(7)العناصر الثمانية في جروب األوتومورفيسم 
𝑃𝑆𝐿2(7)  2.المجموعة

𝑒𝑖±وتحافظ على مجموعة الكوتنيونس  𝐺2هي مجموعة جزئية منتهية من ليا جروب  3 , (𝑖 = 1,2, … وتمثل أوكتاهدرون   (7,
:23  في سبعة أبعاد. لها ثالثة مجموعة جزئية  7: .23 و 3 𝑆4 4و. 𝑆4: و  𝐺2وهذه المجموعات الجزئية تمثل مجموعات جزئية منتهية من ليا جروب  2

𝑆𝑂(4)   و𝑆𝑈(3)  23بالترتيب. المجموعة: 𝑃𝑆𝐿2 2 تمثل االمتداد المقسوم تمتلك خمس مجموعات جزئية وهي  (7)
3: 7: :23 و  3 𝑆4 4و: 𝑆4: 2 

ول الخصائص لهذه المجموعات ومجموعاتهن الجزئية ومنتجات التنسر وتقسيماتهن . جدا𝑃𝑆𝐿2(7)واثنتين من المجموعات الغير المرافقة للكلين جروب 
 إلى تمثيالتهن الريديوسبل تحت مجموعاتهن الجزئية كلها تم تحديدها. كما تمت مناقشة بعض التطبيقات في الفيزياء.

 

 .الخصائص، منتجات التنسر المجموعات المنتهية، كواتنينس المنفصلة، إمتداد المجوعة، جدول: مفتاحيةالكلمات ال

1. Introduction 

e have introduced some part of this work in an earlier publication [1]. Since then we have observed that the 

simple group like 𝑃𝑆𝐿2(7) [2] and the subgroups thereof  7: 3 [3], 𝑆4 and 𝐴4 [4, 5, 6, 7, 8, 9] have been 
proposed to explain the properties of the Tri-Bimaximal Neutrino Mixing [10]. When the charged leptons and quark 

masses are incorporated into the scheme we may think of much larger discrete symmetries broken down to the 

aforementioned finite subgroups of 𝑆𝑈(3). Along with these lines we would like to introduce two groups of order 1344 
which are non-split and split extensions of the elementary Abelian group  23 of order 8 by its automorphism group 
𝑃𝑆𝐿2(7). Before we proceed further a glossary may be introduced for the group theoretical concepts and notations 
used throughout the paper. We follow the notations of the Atlas of Finite Groups [11]. A cyclic group of order p is 

denoted by p. An elementary Abelian group of order 𝑝𝑛  (denoted also by 𝑝𝑛 ) is the direct product of n cyclic groups of 
each having order p. Thus, the elementary abelian group  23 is the direct product of three cyclic groups of order 2.  
 

 

 
 

W 



MEHMET KOCA  ET AL 

 

58 

 

The group A.B denotes any group possessing a normal subgroup A, for which the corresponding quotient group is 

B. This is used for the non-split extension. The group A:B indicates the split extension, or the  semi-direct product 𝐴 ⋊
𝐵. Here a copy of the quotient group B is a subgroup of the group A:B. This shows that the intersection of A and B is 
just the unit element. The quotient group of interest here is the special projective group 𝑃𝑆𝐿2(7) and this has been 
discussed in the physics literature extensively [2,12,13]. 

The paper is organized as follows. For the group  23.𝑃𝑆𝐿2(7) which is the automorphism group of the octonionic 
set ±𝑒𝑖, (𝑖 = 1,2, … ,7) we review in Section 2 the basic structure of the octonion algebra and introduce the group 7: 3 
as the automorphism of the 7 imaginary units 𝑒𝑖, (𝑖 = 1,2, … ,7). In Section 3 we extend the automorphism to the full 
group of automorphism including the change of sign of the imaginary units of octonions and point out that the diagonal 

matrices form the elementary abelian subgroup of the automorphism group. The quotient group 𝑃𝑆𝐿2(7) is explicitly 
constructed and the maximal subgroups of the group  23.𝑃𝑆𝐿2 (7) are identified. In Section 4 we discuss the 
construction of the group by using a finite subgroup of  𝑆𝑂(4) based on the preservation of the quaternion subalgebra 
of the octonion algebra. Section 5 deals with the construction of the 7-dimensional irreducible representation of the 

group 23: 𝑃𝑆𝐿2(7)  and its five maximal subgroups, two of which, are the non-conjugate Klein’s group 𝑃𝑆𝐿2(7). In the 
concluding Section 6, we point out as to how these groups can be used in physics. In Appendix A we study the tensor 

products of the irreducible representations. Appendix B lists the decompositions of the irreducible representations 

under the maximal subgroups.  

2. Octonions and the group 𝟕: 𝟑 

The octonions (Cayley numbers) are sets of real numbers 

 

                                                  𝑞 = (𝑞0, 𝑞1, 𝑞2, 𝑞3, 𝑞4, 𝑞5, 𝑞6, 𝑞7) 

= 𝑞0. 1 + 𝑞1𝑒1 + 𝑞2𝑒2 + 𝑞3𝑒3 + 𝑞4𝑒4 + 𝑞5𝑒5 + 𝑞6𝑒6 + 𝑞7𝑒7                                    (1) 
 

where 𝑒𝑖 , (𝑖 = 1,2, … . ,7) are 7 imaginary octonionic units. Octonions are added like vectors and multiplied as follows 
 

1. 𝑒𝑖 = 𝑒𝑖 . 1 = 𝑒𝑖 , 

𝑒𝑖 𝑒𝑗 = −𝛿𝑖𝑗 + ∑ 𝜙𝑖𝑗𝑘
7
𝑘=1 𝑒𝑘     (𝑖, 𝑗, 𝑘 = 1,2, … ,7)                                             (2) 

 

where 𝜙𝑖𝑗𝑘 are completely antisymmetric in 𝑖, 𝑗, 𝑘 with the values ±1. We choose the basis as shown in Figure 1 such 

that [14] 

 

𝜙123 = 𝜙246 = 𝜙435 = 𝜙367 = 𝜙651 = 𝜙572 = 𝜙714 = 1,                                     (3) 

 

which follows from the cyclic rotation of the triangle in Figure 1. 

 

 

 

 

 

 

Figure 1. Octonionic multiplication based on quaternionic multiplication. 

 

 

The 7-imaginary units form 35 triads, 7 of which are associative and follow the ordering such that 𝑒𝑖 𝑒𝑗 𝑒𝑘 = −1 when 

𝑖, 𝑗, 𝑘 take one of the values in (3). 
 The 28 anti-associative triads can be obtained from the following four anti-associative triads 

 

       𝑒1(𝑒2𝑒4) = 𝑒5 = −(𝑒1𝑒2)𝑒4 
−𝑒1(𝑒2𝑒6) = 𝑒7 = (𝑒1𝑒2)𝑒6 
−𝑒1(𝑒2𝑒5) = 𝑒4 = (𝑒1𝑒2)𝑒5 

𝑒1(𝑒2𝑒7) = 𝑒6 = −(𝑒1𝑒2)𝑒7                                                                 (4) 

𝑒5 

𝑒1 

𝑒2 

𝑒4 

𝑒3 𝑒6 

𝑒7 



TWO GROUPS 𝟐𝟑.𝑷𝑺𝑳𝟐(𝟕) AND 𝟐
𝟑: 𝑷𝑺𝑳𝟐(𝟕) OF ORDER 1344 

 

59 

 

by using the cyclic permutation (1243657). The seven associative triads correspond to the lines of the finite projective 
geometry of seven lines and seven points of the Fano plane. 

Without changing the signs of the octonionic imaginary units in Figure 1, we can obtain two operations preserving the 

octonion algebra.  Let 𝛼 transform the octonionic units 𝑒𝑖  in the cyclic order (𝑒1𝑒2𝑒4𝑒3𝑒6𝑒5𝑒7) so that 𝛼
7 = 1. Let 𝛽 

fix 𝑒1 and permute the associative triad (𝑒2𝑒4𝑒6) and the anti-associative triad (𝑒3𝑒7𝑒5)  in the indicated orders. It is 

clear that 𝛽3 = 1 and preserves the octonion algebra. Indeed, the transformation (𝑒3𝑒7𝑒5)  is sufficient to determine the 

rest of the transformations of the quaternionic imaginary units. It is easy to prove that two generators generate a finite 

group of order 21 satisfying the generation relations: 

 

                                           𝛼 7 = 𝛽3 = 𝛽−1 𝛼𝛽𝛼 3 = 1.                                                                     (5) 
 

It is clear from (5) that 𝛼 generates a normal subgroup of order 7 and hence the structure of the group 7: 3. The 
conjugacy classes are given by the set of group elements 

 

𝐶1 = (1),   𝐶2 = ( 𝛼, 𝛼
2, 𝛼 4), 𝐶3 = ( 𝛼

3, 𝛼 5, 𝛼 6), 

         𝐶4 = (𝛼
𝑎 𝛽), 𝐶5 = (𝛼

𝑎𝛽2), (𝑎 = 0, 1, … , 6).                                                       (6)   
 

The character table of the group is depicted in Table 1. 𝐶𝑖
[𝑗]

 denotes the i
th 

conjugacy class and [j] represents the order 

of the elements in the conjugacy class. 𝜒[𝑎] is the character of the irreducible representation 𝑎. 
 

 

Table 1. The characteristics table of the Frobenius group 7: 3. 

7: 3 𝐶1 7𝐶2
[3]

 7𝐶3
[3]

 3𝐶4
[7]

 3𝐶5
[7]

 

𝜒[1] 1 1 1 1 1 

𝜒[11] 1 𝜇 𝜇 1 1 

𝜒[12] 1 𝜇 𝜇 1 1 

𝜒[31] 3 0 0 𝜂 𝜂 

𝜒[32] 3 0 0 𝜂 𝜂 

 

Here,  𝜒[12] = 𝜒[ 11 ], 𝜒[32] = 𝜒[  31], 𝜇 =
1

2
(−1 + 𝑖√3), 𝜂 =

1

2
(−1 + 𝑖√7). 

 

One can easily check that the 7-dimensional representation obtained by the generators 𝛼 and 𝛽 is reducible and can be 

decomposed as 7 = 1 + 31 + 31. 

3. The group 𝟐𝟑.𝑷𝑺𝑳𝟐(𝟕) as the automorphism group of the octonionic set ±𝒆𝒊 

Now we allow the octonionic units 𝑒𝑖 to also take negative values, or, in other words, we also include the 

octonionic conjugates in the transformations. Any linear transformation on a non-associative triad determines the 

transformations of 7 imaginary units. Let us take the non-associative triad (𝑒1𝑒2𝑒7) and impose the transformation 

𝑒1 → 𝑒1, 𝑒2 → 𝑒2, 𝑒7 → −𝑒7. This transformation leads to the transformations 𝑒3 → 𝑒3, 𝑒4 → −𝑒4, 𝑒5 → −𝑒5 and 

𝑒6 → −𝑒6. Let us call it 𝑁1 = (𝑒1 → 𝑒1, 𝑒2 → 𝑒2, 𝑒3 → 𝑒3, 𝑒4 → −𝑒4, 𝑒5 → −𝑒5, 𝑒6 → −𝑒6, 𝑒7 → −𝑒7)   and denote it 

as a diagonal matrix with non-zero entities 

 

𝑁1 = (1, 1, 1, −1, −1, −1, −1).                                                                   (7) 

 

This diagonal transformation leaves the associative triad (123) intact and changes the signs of the other 4 octonionic 

units. When we take 𝛼, 𝛽 and 𝑁1 as generators we obtain a group of order 168, but not isomorphic to 𝑃𝑆𝐿2(7). It has, 

by construction, a normal subgroup of order 8 generated by three diagonal transformations 𝑁1, 𝑁2 and 𝑁7 where 

𝑁2 = (1, −1, −1, 1, −1, −1, 1) and 𝑁7 = (−1, 1, −1, 1, −1,1, −1). They generate the elementary Abelian group 2
3 

where the other elements are defined as 

 

𝑁3 = 𝑁1𝑁2 = 𝑁2𝑁1, 𝑁4 = 𝑁7𝑁1 = 𝑁1𝑁7, 

  𝑁5 = 𝑁7𝑁2 = 𝑁2𝑁7, 𝑁6 = 𝑁7𝑁3 = 𝑁3𝑁7.                                                         (8)                                                                                                         
 

 



MEHMET KOCA  ET AL 

 

60 

 

More compactly, they can be written as  

 
𝑁𝑖 𝑁𝑗 = 𝑁𝑗 𝑁𝑖 = 𝑁𝑘 , (𝑖𝑗𝑘 = 123, 147, 165, 246, 257, 345, 367).                                       (9) 

 
The 7-diagonal matrices can be used to define the Fano plane. One can show that the set of group elements 𝑁𝑖 , (𝑖 =
1,2, … ,7) is invariant under the conjugations of the generators so that the group can be designated as  23: 7: 3 the order 
of which is 168 but since it has a normal subgroup it is not isomorphic to the simple group 𝑃𝑆𝐿2 (7). The 7-
dimensional representation of the group  23: 7: 3 obtained from the generators 𝛼, 𝛽 and 𝑁1is irreducible and denoted 
by 71 in the character table of the group  2

3: 7: 3 given in Table 2.  

Table 2. The characteristics table of the group  23: 7: 3. 

23: 7: 3 𝐶1 7𝐶2
[2]

 28𝐶3
[3]

 28𝐶4
[3]

 28𝐶5
[6]

 28𝐶6
[6]

 24𝐶7
[7]

 24𝐶8
[7]

 

𝜒[1] 1 1 1 1 1 1 1 1 

𝜒[11] 1 1 𝜇 𝜇 𝜇 𝜇 1 1 

𝜒[12] 1 1 𝜇 𝜇 𝜇 𝜇 1 1 

𝜒[31] 3 3 0 0 0 0 𝜂 𝜂 

𝜒[32] 3 3 0 0 0 0 𝜂 𝜂 

𝜒[71] 7 -1 1 1 -1 -1 0 0 

𝜒[72] 7 -1 𝜇 𝜇 −𝜇 −𝜇 0 0 

𝜒[73] 7 -1 𝜇 𝜇 −𝜇 −𝜇 0 0 

 

Note that in Table 2 the characters satisfy the relations𝜒[12] = 𝜒[ 11 ], 𝜒[32] = 𝜒[ 31 ],  

𝜒[73] = 𝜒[ 71 ] and the other irreducible representations are real. The group  23: 7: 3 is not the full automorphism group 
of the set of octonions ±𝑒𝑖  . In fact, a transformation of the form 𝛾: ( 𝑒1 ↔ −𝑒4, 𝑒2 ↔ −𝑒5, 𝑒3 → −𝑒3, 𝑒6 → 𝑒6, 𝑒7 →
−𝑒7) preserves the octonion algebra and does not belong to the group of elements of the group  2

3: 7: 3. Adjoining 𝛾 as 
a new generator then 𝛼, 𝛽, 𝛾 and 𝑁1 together generate the group of order 1344. We will prove that it has the structure 
 23.𝑃𝑆𝐿2(7) which can also be generated by two generators 𝛼 and 𝛾 only. Here the group 𝑃𝑆𝐿2(7) is the quotient 
group whose generators are defined by the conjugations over the elements of the elementary abelian group 23: 

 
𝛼 ̃: 𝑁𝑖 → 𝛼

−1𝑁𝑖 𝛼,  𝛽 ̃: 𝑁𝑖 → 𝛽
−1𝑁𝑖 𝛽, 𝛾 ̃: 𝑁𝑖 → 𝛾

−1𝑁𝑖 𝛾.                                          (10) 
They permute the diagonal matrices as  

 

𝛼 ̃ = (𝑁1𝑁2𝑁4𝑁3𝑁6𝑁5𝑁7), 

      𝛽 ̃ = (𝑁3𝑁2𝑁1)(𝑁4𝑁6𝑁5)(𝑁7) 

 𝛾 ̃ = (𝑁1𝑁5)(𝑁2)(𝑁3𝑁7)(𝑁4)(𝑁6).                                                       (11) 

 

The group generated by the generators in (11) is isomorphic to the Klein’s group 𝑃𝑆𝐿2 (7) of order 168 and the 7-
dimensional representation obtained from (11) is reducible 7 = 1 + 6. This is expected because 7 diagonal matrices 
𝑁𝑖  form the Fano plane whose automorphism group is the Klein’s group. The character table of the group 𝑃𝑆𝐿2(7) is 
shown in Table 3. (Note the difference between Table 2 and Table 3). 

Table 3. The characteristics table of the group 𝑃𝑆𝐿2(7). 

𝑃𝑆𝐿2(7) 𝐶1 21𝐶2
[2]

 56𝐶3
[3]

 42𝐶4
[4]

 42𝐶5
[7]

 42𝐶6
[7]

 

𝜒[1] 1 1 1 1 1 1 

𝜒[31] 3 -1 0 1 𝜂 𝜂 

𝜒[32] 3 -1 0 1 𝜂 𝜂 

𝜒[6] 6 2 0 0 -1 -1 

𝜒[7] 7 -1 1 -1 0 0 

𝜒[8] 8 0 -1 0 1 1 

 

Here again the characters satisfy 𝜒[32] = 𝜒[31]. A copy of the 𝑃𝑆𝐿2(7) does not exist in the group  2
3.𝑃𝑆𝐿2 (7), 

therefore it is not a subgroup and hence the latter group is the non-split extension of the group  23 by the group 
𝑃𝑆𝐿2(7). The character table of the groups  2

3.𝑃𝑆𝐿2(7) and  2
3: 𝑃𝑆𝐿2(7) is given in Table 4. 



TWO GROUPS 𝟐𝟑.𝑷𝑺𝑳𝟐(𝟕) AND 𝟐
𝟑: 𝑷𝑺𝑳𝟐(𝟕) OF ORDER 1344 

 

61 

 

Table 4. The characteristics table of the groups 23.𝑃𝑆𝐿2(7) and  2
3: 𝑃𝑆𝐿2(7). 

 
23. 𝑃𝑆𝐿2(7) 𝐶1 𝐶2

[2]
 𝐶3

[2]
 𝐶4

[3]
 𝐶5

[4]
 𝐶6

[4]
 𝐶7

[6]
 𝐶8

[7]
 𝐶9

[7]
 𝐶10

[8]
 𝐶11

[8]
 

23: 𝑃𝑆𝐿2(7) 𝐶1 𝐶2
[2]

 𝐶3
[4]

 𝐶4
[3]

 𝐶5
[2]

 𝐶6
[2]

 𝐶7
[6]

 𝐶8
[7]

 𝐶9
[7]

 𝐶10
[4]

 𝐶11
[4]

 

Elements 1 7 84 224 42 42 224 192 192 168 168 

𝜒[1] 1 1 1 1 1 1 1 1 1 1 1 

𝜒[31] 3 3 -1 0 -1 -1 0 𝜂 𝜂 1 1 

𝜒[32] 3 3 -1 0 -1 -1 0 𝜂 𝜂 1 1 

𝜒[6] 6 6 2 0 2 2 0 -1 -1 0 0 

𝜒[71] 7 -1 -1 1 -1 3 -1 0 0 1 -1 

𝜒[72] 7 7 -1 1 -1 -1 1 0 0 -1 -1 

𝜒[73] 7 -1 -1 1 3 -1 -1 0 0 -1 1 

𝜒[8] 8 8 0 -1 0 0 -1 1 1 0 0 

𝜒[14] 14 -2 -2 -1 2 2 1 0 0 0 0 

𝜒[211] 21 -3 1 0 -3 1 0 0 0 -1 1 

𝜒[212] 21 -3 1 0 1 -3 0 0 0 1 -1 

 
The character table shows that all the representations are real except the 3-dimensional representations which 

satisfy 𝜒[32] = 𝜒 [31 ]. The group 23.𝑃𝑆𝐿2(7) has three maximal subgroups having the structures  2
3: 7: 3, 

 4. 𝑆4: 2  and 2
3. 𝑆4 of respective orders 168, 192 and 192. The group  2

3: 7: 3 preserves the octonion multiplication and 
the 7-dimensional representation denoted by 71 is irreducible. It is a subgroup of the Lie group 𝐺2 and the groups 
 4. 𝑆4: 2  and 2

3. 𝑆4 are   finite subgroups of the groups 𝑆𝑈(3) and 𝑆𝑂(4) respectively, as we will study in the next 
section. The Lie groups 𝑆𝑈(3) and 𝑆𝑂(4) are the maximal subgroups of the Lie group 𝐺2.  

3.1  Maximal subgroups 𝟒. 𝑺𝟒: 𝟐   𝐚𝐧𝐝 𝟐
𝟑. 𝑺𝟒 of the group 𝟐

𝟑.𝑷𝑺𝑳𝟐(𝟕) 

The group  23.𝑃𝑆𝐿2 (7) has two maximal subgroups of order 192. Now we discuss some properties of the first 
subgroup 4. 𝑆4: 2. Let 𝑒𝑖 = −𝑒𝑖  denote the octonionic conjugate. Define the transformation  
 

𝜃 = (𝑒1𝑒5)(𝑒2𝑒3𝑒4𝑒7𝑒2𝑒3𝑒4𝑒7)(𝑒6𝑒6),    with 𝜃
8=1.                                              (12) 

 

The generators 𝛾 and 𝜃 generate the group 4. 𝑆4: 2 which represents a finite subgroup of  𝑆𝑈(3) as the generators 
fix the octonion ±𝑒6. The character table of the group 4. 𝑆4: 2 is shown in Table 5.  

 

Table 5. The characteristics table of the groups 4. 𝑆4: 2 and 4: 𝑆4: 2 

4. 𝑆4: 2 𝐶1 𝐶2
[2]

 𝐶3
[2]

 𝐶4
[2]

 𝐶5
[2]

 𝐶6
[2]

 𝐶7
[3]

 𝐶8
[4]

 𝐶9
[4]

 𝐶10
[4]

 𝐶11
[4]

 𝐶12
[6]

 𝐶13
[8]

 𝐶14
[8]

 

4: 𝑆4: 2 𝐶1 𝐶2
[2]

 𝐶3
[2]

 𝐶4
[4]

 𝐶5
[4]

 𝐶6
[4]

 𝐶7
[3]

 𝐶8
[2]

 𝐶9
[2]

 𝐶10
[2]

 𝐶11
[2]

 𝐶12
[6]

 𝐶13
[4]

 𝐶14
[4]

 

Elements 1 3 4 12 12 12 32 12 6 6 12 32 24 24 

𝜒[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

𝜒[11] 1 1 -1 1 -1 -1 1 1 1 1 -1 -1 1 -1 

𝜒[12] 1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 1 

𝜒[13] 1 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 

𝜒[21] 2 2 -2 0 0 -2 -1 0 2 2 0 1 0 0 

𝜒[22] 2 2 2 0 0 2 -1 0 2 2 0 -1 0 0 

𝜒[31] 3 3 3 -1 -1 -1 0 -1 -1 -1 -1 0 1 1 

𝜒[32] 3 3 3 1 1 -1 0 1 -1 -1 1 0 -1 -1 

𝜒[33] 3 3 -3 1 -1 1 0 1 -1 -1 -1 0 -1 1 

𝜒[34] 3 3 -3 -1 1 1 0 -1 -1 -1 1 0 1 -1 

𝜒[61] 6 -2 0 0 -2 0 0 0 -2 2 2 0 0 0 

𝜒[62] 6 -2 0 0 2 0 0 0 -2 2 -2 0 0 0 

𝜒[63] 6 -2 0 -2 0 0 0 2 2 -2 0 0 0 0 

𝜒[64] 6 -2 0 2 0 0 0 -2 2 -2 0 0 0 0 



MEHMET KOCA  ET AL 

 

62 

 

This is also one of the maximal subgroups of the Chevalley’s group 𝐺2(2) [15] preserving the octonionic root 
system of 𝐸7. The 7-dimensional representation can be written as 7 = 1 + 6, or more properly, 71 = 11 + 63. By 
adjoining the generator 𝛼 which permutes the 7 imaginary octonionic units one can generate the group  23.𝑃𝑆𝐿2(7). It 
is easy to prove that  𝛾 ̃and �̃� = (𝑁1𝑁4𝑁2𝑁5)(𝑁3)(𝑁6𝑁7) with 𝛾 ̃

2 = �̃�4 = 1 generate the subgroup 𝑆4 as the quotient 
group. Let 𝑎 and 𝑏 be the generators of a group, then the standard generation relation of 𝑆4 is 𝑎

4 = 𝑏3 = (𝑎𝑏)2 = 1 
[16]. This relation can be satisfied if we define 𝑎 = 𝛾 ̃�̃�𝛾 ̃ and 𝑏 = 𝛾 ̃ �̃�−1.   
 

The group   23. 𝑆4 can be generated  by the generators  

 
𝐴 = (𝑒1𝑒7 𝑒3𝑒1𝑒7𝑒3)(𝑒2𝑒4𝑒6𝑒2𝑒4𝑒6)( 𝑒5𝑒5), 

𝐵 = (𝑒1) (𝑒2𝑒6𝑒2𝑒6)( 𝑒3𝑒5𝑒3𝑒5)(𝑒4)(𝑒7),                                                          (13) 

                                                           𝐴6 = 𝐵4 = 1.   

 

The elements  𝐴 ̃and  𝐵 ̃can be obtained as follows 
 

𝐴 ̃ = (𝑁1𝑁6𝑁2)(𝑁3𝑁5𝑁4)(𝑁7), 𝐵 ̃ = (𝑁1𝑁3)(𝑁2)(𝑁4𝑁6)(𝑁5)(𝑁7).                                   (14) 
 

Now define 𝑎 = 𝐵 ̃𝐴 ̃ = (𝑁1𝑁4)(𝑁2𝑁3𝑁5𝑁6)(𝑁7) and 𝑏 = �̃�
−1.  It is straightforward to show that they satisfy the 

𝑆4 generation relation 𝑎
4 = 𝑏3 = (𝑎𝑏)2 = 1.  Hence, the quotient group is 𝑆4 as claimed. The matrix representations 

of 𝐴 ̃and 𝐵 ̃do not preserve the octonion algebra, and therefore a copy of 𝑆4 does not exist in the group  2
3. 𝑆4. Hence it 

is the non-split extension of the group  23 by 𝑆4. 
It is clear from the generators 𝐴 and 𝐵  that they preserve the sets of octonionic units (±𝑒2 ± 𝑒4 ± 𝑒6) and (±𝑒1 ±
𝑒3 ± 𝑒5 ± 𝑒7) separately, so that the 7 × 7 matrix representation can be written in the block diagonal form of 3 ×
3  and 4 × 4 matrices. This proves that the 7-dimensional representation of the group  23.𝑃𝑆𝐿2(7) can be decomposed 
as 71 = 31 + 41 under the group  2

3. 𝑆4. 
A rigorous proof can also be given by noting that the group  23. 𝑆4 is a finite subgroup of 𝑆𝑂(4) which preserves the 
quaternion substructure of the octonion algebra.  

Let h represent a quaternion. Then an octonion can be written as ℎ + 𝑒7ℎ where the quaternionic imaginary units are 
taken as 𝑒1, 𝑒2 and 𝑒3. It can be proved that the quaternion preserving transformation on the octonion ℎ + 𝑒7ℎ [17] can 
be written as  

ℎ + 𝑒7ℎ → 𝑝ℎ𝑝 + 𝑝ℎ𝑞                                                                        (15) 
 

where p and q are unit quaternions. The first term 𝑝ℎ𝑝 preserves the scalar part of the octonion. Therefore, without loss 
of generality, we can also write it as 𝑝  𝐼𝑚(ℎ)𝑝. Since we are interested in the transformations of the set ±𝑒𝑖 , (𝑖 =
1,2, … ,7) we can write it as  

𝐼𝑚 (𝑉0) + 𝑒7𝑉0 → 𝑝𝐼𝑚( 𝑉0)𝑝 + 𝑒7𝑝𝑉0𝑞                                                         (16)      
 

where the set 𝑉0 = {±1, ±𝑒1, ±𝑒2, ±𝑒3} represents the elements of the quaternion group of order 8. So, the question 
now turns out to be for which unit quaternions p and q the set of quaternions 𝑉0 is preserved. The answer lies in the 
finite quaternion subgroups involving  𝑉0 as a subgroup. They are the binary tetrahedral group 𝒯, binary octahedral 
group 𝒪 and the binary icosahedral group ℐ[16]. It turns out that we need to invoke the binary octahedral group whose 
sets of elements can be written as the union of 6 quaternionic sets [18] 

 

      𝒪 = 𝑉0 + 𝑉+ + 𝑉− + 𝑉1 + 𝑉2 + 𝑉3.                                                          (17) 
Here the sets of quaternions are defined as  

𝑉0 = {±1, ±𝑒1, ±𝑒2, ±𝑒3} 

𝑉+ =
1

2
(±1 ± 𝑒1 ±𝑒2 ± 𝑒3), even number of (+sign) 

𝑉− =
1

2
(±1 ± 𝑒1 ±𝑒2 ± 𝑒3), odd number of (+sign) 

𝑉1 = {
1

√2
(±1 ± 𝑒1 ),

1

√2
( ±𝑒2 ± 𝑒3)},                                                                       (18) 

                                                              𝑉2 = {
1

√2
(±1 ± 𝑒2 ),

1

√2
( ±𝑒3 ± 𝑒1)},                                                        𝑉3 =

{
1

√2
(±1 ± 𝑒3 ),

1

√2
( ±𝑒1 ± 𝑒2)}. 

 
They satisfy the multiplication table as shown in Table 6.  

 

 

 

 



TWO GROUPS 𝟐𝟑.𝑷𝑺𝑳𝟐(𝟕) AND 𝟐
𝟑: 𝑷𝑺𝑳𝟐(𝟕) OF ORDER 1344 

 

63 

 

Table 6. Multiplication table for the binary octahedral group. 

 𝑉0 𝑉+ 𝑉− 𝑉1 𝑉2 𝑉3 

𝑉0 𝑉0 𝑉+ 𝑉− 𝑉1 𝑉2 𝑉3 

𝑉+ 𝑉+ 𝑉− 𝑉0 𝑉3 𝑉1 𝑉2 

𝑉− 𝑉− 𝑉0 𝑉+ 𝑉2 𝑉3 𝑉1 

𝑉1 𝑉1 𝑉2 𝑉3 𝑉0 𝑉+ 𝑉− 

𝑉2 𝑉2 𝑉3 𝑉1 𝑉− 𝑉0 𝑉+ 

𝑉3 𝑉3 𝑉1 𝑉2 𝑉+ 𝑉− 𝑉0 

 
To determine the pair of quaternions p and q we use the notation 𝑝ℎ𝑞 ≔ [𝑝, 𝑞]ℎ. Later, we simply drop the quaternion 
h and write the 𝑆𝑂(4) group elements as the pair [𝑝, 𝑞]. Since we have 
 

𝑉0 = 𝑝𝑉0𝑞 → 𝑞 = 𝑉0𝑝𝑉0 = 𝑉0𝑝𝑉0                                                           (19) 
 

and p can take values from the sets 𝑉0, 𝑉+, 𝑉−, 𝑉1, 𝑉2, 𝑉3, then the corresponding sets of quaternions 𝑞 can be 
determined from the Table 6 as 𝑉0, 𝑉−, 𝑉+, 𝑉1, 𝑉2, 𝑉3. Therefore the 𝑆𝑂(4) group elements preserving the set  𝑉0 can be 
written as the union of pairs of sets 

 
[𝑉0, 𝑉0] + [𝑉+, 𝑉−] + [𝑉−, 𝑉+] + [𝑉1, 𝑉1] + [𝑉2, 𝑉2] + [𝑉3, 𝑉3].                                (20) 

 

This is a group of order 192 with 13 conjugacy classes which can be converted to a 4 dimensional irreducible 

representation if the basis is chosen as the unit quaternions 1, 𝑒1, 𝑒2, 𝑒3 . When the term [𝑝, 𝑝] is incorporated into the 
group  23. 𝑆4, it can be converted into a 7 × 7 matrix representation in the block diagonal form of 3 × 3  and 4 × 4 
matrices. This proves that the 7-dimensional irreducible representation of the group  23.𝑃𝑆𝐿2 (7) can be decomposed as 
71 = 31 + 41 under the irreducible representations of the group 2

3. 𝑆4 . The character table of the group  2
3. 𝑆4 is given 

in Table 7. Its elementary Abelian subgroup  23 can be generated by the group elements, say [1, −1], [𝑒1, −𝑒1] and [𝑒2, 
−𝑒2,], and it can be proven that the group  2

3 is a normal subgroup. It is also easy to identify its maximal subgroups. 

For example, the set [𝑉0, 𝑉0] + [𝑉+, 𝑉−] + [𝑉−, 𝑉+] forms a group of order 96 and [𝑉0, 𝑉0] + [𝑉1, 𝑉1] forms a group of 
order 64. 

Table 7. The characteristics table of the groups  23. 𝑆4 and  2
3: 𝑆4 

 

 23. 𝑆4 𝐶1 𝐶2
[2]

 𝐶3
[2]

 𝐶4
[2]

 𝐶5
[2]

 𝐶6
[3]

 𝐶7
[4]

 𝐶8
[4]

 𝐶9
[4]

 𝐶10
[4]

 𝐶11
[6]

 𝐶12
[8]

 𝐶13
[8]

 

 23: 𝑆4 𝐶1 𝐶2
[2]

 𝐶3
[2]

 𝐶4
[4]

 𝐶5
[4]

 𝐶6
[3]

 𝐶7
[2]

 𝐶8
[2]

 𝐶9
[2]

 𝐶10
[2]

 𝐶11
[6]

 𝐶12
[4]

 𝐶13
[4]

 

Elements 1 1 6 12 24 32 6 6 12 12 32 24 24 

𝜒[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 

𝜒[11] 1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 

𝜒[2] 2 2 2 2 0 -1 2 2 0 0 -1 0 0 

𝜒[31] 3 3 -1 -1 -1 0 -1 3 1 1 0 -1 1 

𝜒[32] 3 3 -1 -1 1 0 -1 3 -1 -1 0 1 -1 

𝜒[33] 3 3 3 -1 1 0 -1 -1 1 1 0 -1 -1 

𝜒[34] 3 3 -1 -1 -1 0 3 -1 1 1 0 1 -1 

𝜒[35] 3 3 -1 -1 1 0 3 -1 -1 -1 0 -1 1 

𝜒[36] 3 3 3 -1 -1 0 -1 -1 -1 -1 0 1 1 

𝜒[41] 4 -4 0 0 0 1 0 0 -2 2 -1 0 0 

𝜒[42] 4 -4 0 0 0 1 0 0 2 -2 -1 0 0 

𝜒[6] 6 6 -2 2 0 0 -2 -2 0 0 0 0 0 

𝜒[8] 8 -8 0 0 0 -1 0 0 0 0 1 0 0 

 

Since the group  23. 𝑆4 leaves the subsets (±𝑒1, ±𝑒2, ±𝑒3) and  𝑒7(±1, ±𝑒1, ±𝑒2, ±𝑒3)    invariant it is easy to construct 
its 7 conjugate groups by permuting the quaternionic subsets (123), (246), (435), ( 367), (652), ( 572), (714) by the 



MEHMET KOCA  ET AL 

 

64 

 

group generator 𝛼 so that the group is extended by the inclusion of the generator 𝛼 to the full automorphism group  
 23.𝑃𝑆𝐿2(7) of the octonionic set ±𝑒𝑖 . 

4. Construction of the 7-dimensional irreducible representation of the group  𝟐𝟑:𝑷𝑺𝑳𝟐(𝟕)  

Although the groups are not isomorphic to each other, the split extension 23:𝑃𝑆𝐿2(7) has the same character 
table with the non-split extension 23. 𝑃𝑆𝐿2(7). A close inspection shows that the class structures are the same although 
the powers of the group elements are not always the same. To give an example from the lower rank groups possessing 

the same character table, we may quote the dihedral group of order 8 and the quaternion group denoted by 𝑉0 in our 
notation. The group 23:𝑃𝑆𝐿2(7)  does not preserve the octonion algebra. Then let us assume that the group 
 23:𝑃𝑆𝐿2(7) acts on a 7-dimensional real vector space with the vector components 𝑥𝑖 (𝑖 = 1,2, … ,7) . Let us assume the 
notation 𝑥𝑖 = − 𝑥𝑖 . 

The construction of this group is easy by using its maximal subgroups. We have already discussed that the 

automorphism group 𝑃𝑆𝐿2(7) of the elementary abelian group  2
3can be generated by 𝛼 ̃, 𝛽 ̃and  𝛾 ̃ in (11) and there 

exists a copy of the 𝑃𝑆𝐿2(7) in the group  2
3:𝑃𝑆𝐿2(7). For the representation of 𝛼 ̃, 𝛽 ̃and  𝛾 ̃ defined over 𝑁𝑖, we can 

simply replace 𝑁𝑖  by  𝑥𝑖  and define the diagonal matrices 𝑁𝑖  as if they are acting on the vector components  𝑥𝑖 . We can 
then simply adjoin an element from the elementary Abelian group  23 to generate the whole group. Therefore, it would 

suffice to take  𝛼 ̃, 𝛽 ̃, 𝛾 ̃ of 𝑃𝑆𝐿2 (7) and  𝑁1  as the generators of the group  2
3:𝑃𝑆𝐿2(7). Although this group does not 

preserve the octonion algebra there exists yet a maximal subgroup  23: 7: 3  generated by 𝛼 ̃, 𝛽 ̃and  𝑁1 preserving the 
octonion algebra, as we discussed in Section 3. It is quite natural to expect that the groups  23: 𝑆4 and  4: 𝑆4: 2  are 
maximal subgroups of order 192 in the group  23:𝑃𝑆𝐿2(7). The group  2

3: 𝑆4 can be generated by 𝐴 ̃, 𝐵 ̃and  𝑁1 and its 
character table is depicted in Table 7. Similarly, the subgroup  4: 𝑆4: 2  can be generated by the generators 𝛾 ̃, �̃� and 𝑁1. 
Its character table is displayed in Table 5. 

The group  23:𝑃𝑆𝐿2(7) is a maximal subgroup of the simple group 𝐴8, the even permutations of the 8 letters [11] 
isomorphic to the maximal rotation subgroup of the Coxeter-Weyl group 𝑊(𝑆𝑈(8)) ≅ 𝑆8.  

We have already listed the maximal subgroups of the group 23:𝑃𝑆𝐿2(7) as the groups 𝑃𝑆𝐿2(7),  2
3: 7: 3,  23: 𝑆4 

and 4: 𝑆4: 2 . There exists yet another 𝑃𝑆𝐿2(7) not conjugate to the one generated by 𝛼 ̃, 𝛽 ̃and  𝛾 ̃ , as Conway proved 
[19].  To see this, let us replace the generator 𝛾 ̃ by an another generator 𝛿 = (𝑥1𝑥5)(𝑥2)(𝑥3𝑥7)(𝑥4)(𝑥6), with 𝛿 ∈

 23:𝑃𝑆𝐿2(7) and   𝛿
2 = 1 instead of 𝛾 ̃. One can show that 𝛼 ̃, 𝛽,̃  𝛿 and 𝑁1 generate the same irreducible representation 

of the group  23:𝑃𝑆𝐿2(7) in which the 𝛼 ̃, 𝛽,̃  𝛿 represent the generators of another group 𝑃𝑆𝐿2 (7). This is a miraculous 

structure, in that the 𝑃𝑆𝐿2 (7) =< 𝛼 ̃, 𝛽,̃  𝛿 > is not conjugate to the group 𝑃𝑆𝐿2(7) =< 𝛼 ̃, 𝛽,̃  𝛾 ̃ > . It is 
straightforward to show that �̃�𝛿 = 𝛿𝛾 ̃ = 𝑁7 . It is also true that 7-dimensional representation of the group 𝑃𝑆𝐿2(7) =

< 𝛼 ̃, 𝛽,̃  𝛿 > is irreducible while the 7- dimensional representation of the group 𝑃𝑆𝐿2 (7) =< 𝛼 ̃, 𝛽,̃  𝛾 ̃ >  is reducible 
with 7 = 1 + 6. Conway attributes this exceptional behavior to the holomorph of an elementary Abelian group to 
exhibit this miraculous feature. 

Therefore, the group    23:𝑃𝑆𝐿2(7) has five maximal subgroups  2
3: 7: 3  ,  23: 𝑆4, 4: 𝑆4: 2 and two nonconjugate 

𝑃𝑆𝐿2(7). 

5. Conclusion 

The main motivation was that the groups 23. 𝑃𝑆𝐿2(7) and  2
3: 𝑃𝑆𝐿2(7) could be used as the broken symmetry of 

the mass matrices of quarks, charged leptons and neutrinos when they are considered as a single mass matrix. The 

particle physics literature is full of examples of models demonstrating that the groups 𝑃𝑆𝐿2(7) and its maximal 
subgroups 7: 3 and 𝑆4 can describe the symmetries of the neutrino mass matrix. 

Here we have studied the constructions of two groups 23. 𝑃𝑆𝐿2(7) and  2
3: 𝑃𝑆𝐿2(7) of order 1344 and given 

their character tables, including those of their maximal subgroups. The simple finite subgroup 𝑃𝑆𝐿2(7) of 𝑆𝑈(3) 
occurs either as a factor group or factor and maximal subgroup in these groups. We have also seen that the extension of 

the Frobenious group 7: 3 by the elementary Abelian group 23 leading to the group 23: 7: 3 of order 168 is a subgroup 
in both groups of order 1344. We have also listed the tensor products of the irreducible representations of the groups 

23. 𝑃𝑆𝐿2(7) and  2
3: 𝑃𝑆𝐿2(7) including those of their maximal subgroups in Appendix A and given the 

decompositions of the irreducible representations with respect to the irreducible representations of the relevant 

maximal subgroups in Appendix B. 

Another possible use of these groups in particle physics may arise as follows. The operators, charge 

conjugation 𝐶, parity 𝑃 and time-reversal 𝑇 generate the elementary Abelian group 23 =< 𝐶, 𝑃, 𝑇 > of order 8 when 

they act on the bilinear Dirac fields 𝜓Γ𝜓 constructed by 16 Γ matrices. The Dirac bilinear forms constitute 7 
eigenvectors of the 7 operators generated by the charge conjugation, parity and the time- reversal operators with the 

eigenvalues ±1 besides the scalar bilinear Dirac field which represents the eigenvector of the unit operator. The 
representation of the generators can be taken as the matrices 𝑁𝑖 (𝑖 = 1,2, … ,7). The elementary Abelian group can be 
extended by its automorphism group 𝑃𝑆𝐿2(7) to either of the groups 2

3. 𝑃𝑆𝐿2(7) or 2
3: 𝑃𝑆𝐿2(7). One can construct an 

effective Hamiltonian as the products of Dirac bilinear fields 𝜓Γ𝜓. Formal properties of parity violation, 𝐶𝑃 violation 



TWO GROUPS 𝟐𝟑.𝑷𝑺𝑳𝟐(𝟕) AND 𝟐
𝟑: 𝑷𝑺𝑳𝟐(𝟕) OF ORDER 1344 

 

65 

 

and even 𝐶𝑃𝑇 violation can be explained as invariances under the subgroups of the quotient group 𝑃𝑆𝐿2(7).  If we 
simply impose the 𝛾 ̃ invariance we can prove that the parity violating weak interaction should take the form of either 

𝑉 − 𝐴 or 𝑉 + 𝐴.  If we impose only  𝛽 ̃invariance we obtain CPT preserving but CP violating interaction. Invariance 
under the generator 𝛼 ̃only leads to the CPT violating terms along with a necessary violation of the Lorentz invariance. 

 APPENDIX A. Tensor products 

First of all, we note that the groups 7: 3, 23: 7: 3, 𝑃𝑆𝐿2(7), 2
3. 𝑃𝑆𝐿2(7) and  2

3: 𝑃𝑆𝐿2(7) all have 3-dimensional 
complex representations. Moreover, the group 23: 7: 3 has three 7-dimensional irreducible representations; one real, 
and two complex representations. The real representation preserves the octonion algebra. Tensor products of the 

irreducible representations of the group 𝑃𝑆𝐿2(7) and its maximal subgroups are listed in the reference [2]; therefore we 
will not reproduce them. The other tensor products are given as follows. 

 

Tensor products of the irreducible representations of the group 23: 7: 3 

 
11 × 11 = 12                    12 × 12 = 11 

11 × 12  = 1                                                    12 × 31 = 31 

11 × 31 = 31 12 × 32 = 32 

11 × 32 = 32 12 × 71 = 72 

11 × 71 = 73 12 × 72 = 73 

11 × 72 = 71 12 × 73 = 71 

11 × 73 = 72  

31 × 31 = 32 + 32 + 31 32 × 32 = 31 + 31 + 32 

31 × 32 = 1 + 11 + 12 + 31 + 32 32 × 71 = 71 + 72 + 73 

31 × 71 = 71 + 72 + 73 32 × 72 = 71 + 72 + 73 

31 × 72 = 71 + 72 + 73 32 × 73 = 71 + 72 + 73 

31 × 73 = 71 + 72 + 73  

71 × 71 = 1 + 31 + 32 + 2(71) +  2(72) + 2(73) 

71 × 72 = 12 + 31 + 32 + 2(71) +  2(72) + 2(73) 

71 × 73 = 11 + 31 + 32 + 2(71) +  2(72) + 2(73) 

72 × 72 = 11 + 31 + 32 + 2(71) +  2(72) + 2(73) 

72 × 73 = 1 + 31 + 32 + 2(71) +  2(72) + 2(73) 

73 × 73 = 12 + 31 + 32 + 2(71) +  2(72) + 2(73) 

 
Tensor products of the irreducible representations of the groups 23. 𝑃𝑆𝐿2(7) and  2

3: 𝑃𝑆𝐿2(7) 

 
31 × 31 = 32 + 6 32 × 32 = 31 + 6 

31 × 32 = 1 + 8 32 × 6 = 31 + 72 + 8 

31 × 6 = 32 + 72 + 8 32 × 71 = 212 

31 × 71 = 212 32 × 72 = 6 + 72 + 8 

31 × 72 = 6 + 72 + 8 32 × 73 = 211 

31 × 72 = 211 32 × 8 = 32 + 6 + 72 + 8 

31 × 8 = 31 + 6 + 72 + 8 32 × 14 = 211 + 212 

31 × 14 = 211 + 212 32 × 211 = 73 + 14 + 211 + 212 

31 × 211 = 73 + 14 + 211+212 32 × 212 = 71 + 14 + 211 + 212 

31 × 212 = 71 + 14 + 211+212  

6 × 6 = 1 + 2(6) + 72 + 2(8) 

6 × 71 = 71 + 14 + 211 

6 × 72 = 31 + 32 + 6 + 2(72) +  2(8) 

6 × 73 = 73 + 14 + 212 

6 × 8 = 31 + 32 + 2(6) + 2(72) +  2(8) 

6 × 14 = 71 +  73 + 2(14) + 211+212 

6 × 211 = 71 + 14 + 3(211 ) + 2(212) 



MEHMET KOCA  ET AL 

 

66 

 

6 × 212 = 73 + 14 + 2(211) + 3(212) 

71 × 71 = 1 + 6 + 71 + 14 + 211 

71 × 72 = 73 + 211 + 212 

71 × 73 = 72 + 211 + 212 

71 × 8 = 14 + 211 + 212 

71 × 14 = 6 + 71 + 8 + 14 + 2(211) + 212 

71 × 211 = 6 + 71 + 72 + 73 + 8 + 2(14) + 2(211) + 2(212) 

71 × 212 = 31 + 32 + 72 + 73 + 8 + 14 + 2(211) + 3(212) 

72 × 72 = 1 + 31 + 32 + 2(6) + 2(72) + 2(8) 

72 × 73 = 71 + 211 + 212 

72 × 8 = 31 + 32 + 2(6) + 2(72) + 2(8) 

72 × 14 = 14 + 2(211) + 2(212 ) 

72 × 211 = 71 + 73 + 2(14) + 2(211) + 3(212) 

72 × 212 = 71 + 73 + 2(14) + 3(211) + 2(212) 

73 × 73 = 1 + 6 + 73 + 14 + 212 

73 × 8 = 14 + 211 + 212 

73 × 14 = 6 + 73 + 8 + 14 + 211 + 2(212) 

73 × 211 = 31 + 32 + 71 + 72 + 8 + 14 + 3(211) + 2(212) 

73 × 212 = 6 + 71 + 72 + 73 + 8 + 2(14) + 2(211) + 2(212) 

8 × 8 = 1 + 31 + 32 + 2(6) + 2(72) + 3(8)  

8 × 14 = 71 + 73 + 14 + 2(211) + 2(212) 

8 × 211 = 71 + 73 + 2(14) + 3(211) + 3(212) 

8 × 212 = 71 + 73 + 2(14) + 3(211) + 3(212) 

14 × 14 = 1 + 2(6) + 71 + 72 + 73 + 8 + 2(14) + 3(211) + 3(212) 

14 × 211 = 31 + 32 + 6 + 2(71) + 2(72) + 73 + 2(8) + 3(14) +  5(211) + 4(212) 

14 × 212 = 31 + 32 + 6 + 71 + 2(72) + 2(73) + 2(8) + 3(14) +  4(211) + 5(212) 

211 × 211 = 1 + 31 + 32 + 3(6) + 2(71) + 2(72) + 3(73) + 3(8) + 5(14) + 6(211) + 7(212) 

211 × 212 = 31 + 32 +  2(6) + 2(71) + 3(72) + 2(73) + 3(8) + 4(14) + 7(211) + 7(212 ) 

212 × 212 = 1 + 31 + 32 + 3 (6) + 3(71) + 2(72) + 2(73) + 3(8) + 5(14) + 7(211) + 6(212). 

 

Tensor products of the irreducible representations of the groups 4. 𝑆4: 2 and  4: 𝑆4: 2 

 
11 × 11 = 1                   12 × 12 = 1                   13 × 13 = 1       

11 × 12 =  13       12 × 13 = 11       13 × 21 = 21 

11 × 13 = 12 12 × 21 = 22 13 × 22 = 22 

11 × 21 = 22 12 × 22 = 21                             13 × 31 = 32 

11 × 22 = 21                  12 × 31 = 33 13 × 32 = 31 

11 × 31 = 34 12 × 32 = 34 13 × 33 = 34 

11 × 32 = 33 12 × 33 = 31 13 × 34 = 33 

11 × 33 = 32 12 × 34 = 32 13 × 61 = 62 

11 × 34 = 31 12 × 61 = 61 13 × 62 = 61 

11 × 61 = 62 12 × 62 = 62 13 × 63 = 64 

11 × 62 = 61 12 × 63 = 64 13 × 64 = 63 

11 × 63 = 63 12 × 64 = 63  

11 × 64 = 64   

21 × 21 = 1 + 13 + 22 22 × 22 = 1 + 13 + 22         

21 × 22 = 11 + 12 + 21   22 × 31 = 31 + 32 

21 × 31 = 33 + 34 22 × 32 = 31 + 32 

21 × 32 = 33 + 34 22 × 33 = 33 + 34 

21 × 33 = 31 + 32 22 × 34 = 33 + 34 

21 × 34 = 31 + 32 22 × 61 = 61 + 62 



TWO GROUPS 𝟐𝟑.𝑷𝑺𝑳𝟐(𝟕) AND 𝟐
𝟑: 𝑷𝑺𝑳𝟐(𝟕) OF ORDER 1344 

 

67 

 

21 × 61 = 61 + 62 22 × 62 = 61 + 62 

21 × 62 = 61 + 62 22 × 63 = 63 + 64 

21 × 63 = 63 + 64 22 × 64 = 63 + 64 

21 × 64 = 63 + 64  

31 × 31 = 1 + 22 + 31 + 32 32 × 32 = 1 + 22 + 31 + 32 

31 × 32 = 13 + 22 + 31 + 32 32 × 33 = 11 + 21 + 33 + 34 

31 × 33 = 12 + 21 + 33 + 34 32 × 34 = 12 + 21 + 33 + 34 

31 × 34 = 11 + 21 + 33 + 34 32 × 61 = 61 + 63 + 64 

31 × 61 = 62 + 63 + 64 32 × 62 = 62 + 63 + 64 

31 × 62 = 61 + 63 + 64 32 × 63 = 61 + 62 + 63 

31 × 63 = 61 + 62 + 64 32 × 64 = 61 + 62 + 64 

31 × 64 = 61 + 62 + 63  

33 × 33 = 1 + 22 + 31 + 32 34 × 34 = 1 + 22 + 31 + 32 

33 × 34 = 13 + 22 + 31 + 32 34 × 61 = 61 + 63 + 64 

33 × 61 = 62 + 63 + 64 34 × 62 = 62 + 63 + 64 

33 × 62 = 61 + 63 + 64 34 × 63 = 61 + 62 + 64 

33 × 63 = 61 + 62 + 63 34 × 64 = 61 + 62 + 63 

33 × 64 = 61 + 62 + 64  

61 × 61 = 1 + 12 + 21 + 22 + 32 + 34 + 61 + 62 + 63 + 64 

61 × 62 = 11+13 + 21 + 22 + 31 + 33 + 61 + 62 + 63 + 64 

61 × 63 = 31 + 32 + 33 + 34 + 61 + 62 + 63 + 64 

61 × 64 = 31 + 32 + 33 + 34 + 61 + 62 + 63 + 64 

62 × 62 = 1 + 12++21 + 22 + 32 + 34 + 61 + 62 + 63 + 64 

62 × 63 = 31 + 32 + 33 + 34+61 + 62 + 63 + 64 

62 × 64 = 31 + 32 + 33 + 34+61 + 62 + 63 + 64 

63 × 63 =  1 + 11 + 21 + 22 + 32 + 33 + 61 + 62 + 63 + 64 

63 × 64 = 12+13 + 21 + 22 + 31 + 34 + 61 + 62 + 63 + 64 

64 × 64 =  1 + 11+21 + 22 + 32 + 33 + 61 + 62 + 63 + 64 

 
Tensor products of the irreducible representations of the groups 23. 𝑆4 and  2

3: 𝑆4 

 
11 × 11 = 1       2 × 2 = 1 + 11 +  2       

11 × 2 = 2  2 × 31 = 31 + 33 

11 × 31 = 33 2 × 32 = 32 + 34 

11 × 32 = 34 2 × 33 = 31 + 33 

11 × 33 = 31 2 × 34 = 32 + 34 

11 × 34 = 32 2 × 35 = 35 + 36 

11 × 35 = 36 2 × 36 = 35 + 36 

11 × 36 = 35 2 × 41 = 8 

11 × 41 = 42 2 × 42 = 8 

11 × 42 = 41 2 × 6 = 6 + 6 

11 × 6 = 6 2 × 8 = 41 + 42 + 8 

11 × 8 = 8  

31 × 31 = 1 +  2 + 31 + 33 32 × 32 = 1 +  2 + 32 + 34 

31 × 32 = 36 + 6 32 × 33 = 35 + 6 

31 × 33 = 11 + 2 + 31 + 33 32 × 34 = 11 + 2 + 32 + 34 

31 × 34 = 35 + 6 32 × 35 = 33 + 6 

31 × 35 = 34 + 6 32 × 36 = 31 + 6 

31 × 36 = 32 + 6 32 × 41 = 42 + 8 

31 × 41 = 41 + 8 32 × 42 = 41 + 8 

31 × 42 = 42 + 8 32 × 6 = 31 + 33 + 35 + 36 + 6 



MEHMET KOCA  ET AL 

 

68 

 

31 × 6 = 32 + 34 + 35 + 36 + 6 32 × 8 = 41 + 42 + 2(8) 

31 × 8 = 41 + 42 + 2(8)  

33 × 33 = 1 + 2 + 31 + 33 34 × 34 =  1 + 2 + 32 + 34 

33 × 34 = 36 + 6 34 × 35 = 31 + 6 

33 × 35 = 32 + 6 34 × 36 = 33 + 6 

33 × 36 = 34 + 6 34 × 41 = 41 + 8 

33 × 41 = 42 + 8 34 × 42 = 42 + 8 

33 × 42 = 41 + 8 34 × 6 = 31 + 33 + 35 + 36 + 6 

33 × 6 = 32 + 34 + 35 + 36 + 6 34 × 8 = 41 + 42 + 2(8) 

33 × 8 = 41 + 42 + 2(8)  

35 × 35 = 1 + 2 + 35 + 36 36 × 36 = 1 + 2 + 35 + 36 

35 × 36 = 11 + 2 + 35 + 36 36 × 41 = 42 + 8 

35 × 41 = 41 + 8 36 × 42 = 41 + 8 

35 × 42 = 42 + 8 36 × 6 = 31 + 32 + 33 + 34 + 6 

35 × 6 = 31 + 32 + 33 + 34 + 6 36 × 8 = 41 + 42 + 2(8) 

35 × 8 = 41 + 42 + 2(8)  

41 × 41 = 1 + 31 + 34 + 35 + 6 

41 × 42 = 11 + 32 + 33 + 36 + 6 

41 × 6 = 41 + 42 + 2(8) 

41 × 8 =  2 + 31 + 32 + 33 + 34 + 35 + 36 + 2(6) 

42 × 42 =  1 + 31 + 34 + 35 + 6 

42 × 6 = 41 + 42 + 2(8) 

42 × 8 =  2 + 31 + 32 + 33 + 34 + 35 + 36 + 2(6) 

6 × 6 = 1 + 11 + 2(2) + 31 + 32 + 33 + 34 + 35 + 36 + 2(6) 

6 × 8 =  2(41) +  2(42) + 4(8) 

8× 8 =  1 + 11 + 2 + 2(31) + 2(32) + 2(33) + 2(34) + 2(35) + 2(36) + 4(6) 

 
APPENDIX B.  Decomposition of the irreducible representations in terms of the irreducible representations of 

the maximal subgroups 

 

Decompositions of the irreducible representations of the groups  23. 𝑃𝑆𝐿2 (7) and  2
3: 𝑃𝑆𝐿2(7) under the 

maximal subgroup  23: 7: 3 

Irreducible representations of  23. 𝑃𝑆𝐿2(7) 

and  23: 𝑃𝑆𝐿2(7) 

Irreducible representations of  23: 7: 3 

1 1 

31 31 

32 32 

6 31 + 32 

71 71 

72 1+31 + 32 

73 71 

8 11 + 12 + 31 + 32 

14 72 + 73 

211 71 + 72 + 73 

212 71 + 72 + 73 

 

 

 

 



TWO GROUPS 𝟐𝟑.𝑷𝑺𝑳𝟐(𝟕) AND 𝟐
𝟑: 𝑷𝑺𝑳𝟐(𝟕) OF ORDER 1344 

 

69 

 

Decompositions of the irreducible representations of the group  23: 𝑃𝑆𝐿2(7) under the maximal subgroup 
𝑃𝑆𝐿2(7) 

Irreducible representations of  23: 𝑃𝑆𝐿2(7) Irreducible representations of 𝑃𝑆𝐿2(7) 

1 1 

31 31 

32 = 31 32 = 31 

6 6 

71 7 

72 7 

73 1 + 6 

8 8 

14 6 + 8 

211 31 + 32 + 7 + 8 

212 6 + 7 + 8 

 

Decompositions of the irreducible representations of the group  23: 7: 3 under the maximal subgroup 7: 3 

Irreducible representations of  23: 7: 3 Irreducible representations of 7: 3 

1 1 

11 11 

12 = 11 12 = 11 

31 31 

32 = 31 32 = 31 

71 1 + 31 + 32 

72 12 + 31 + 32 

73 11 + 31 + 32 

 

Decompositions of the irreducible representations of the group 𝑃𝑆𝐿2(7) under the maximal subgroup 7: 3 

Irreducible representations of 𝑃𝑆𝐿2 (7) Irreducible representations of 7: 3 

1 1 

31 31 

32 = 31 32 = 31 

6 31 + 32 

7 1 + 31 + 32 

8 11 + 12 + 31 + 32 

 

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Received   1 May 2018          

Accepted   15 October 2018