EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 3, 2020, 608-619
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On the symmetric block design with parameters
(210,77,28) admitting a Frobenius group of order 57

Menderes Gashi

Department of Mathematics, Faculty of Mathematics and Natural Sciences,
University of Prishtina, Avenue “Mother Teresa” 5, 10000 Prishtina, Kosovo

Abstract. In this paper we have proved that for a putative symmetric block design D with
parameters (210,77,28), admitting a Frobenius group G = 〈ρ, µ|ρ19 = µ3 = 1, ρµ = ρ7〉 of order
57, there are exactly six possible orbit structures up to isomorphism with the orbit distribution
[1; 19; 19; 19; 19; 19; 19; 19; 19; 19; 19; 19].

2020 Mathematics Subject Classifications: 05B05

Key Words and Phrases: Symmetric block design, Orbit structure, Automorphism group

1. Introduction and preliminaries

A 2 − (v, k, λ) design (P,B, I) is said to be symmetric if the relation |P| = |B| = v
holds and in that case we often speak of a symmetric design with parameters (v, k, λ).
The collection of the parameter sets (v, k, λ) for which a symmetric 2 − (v, k, λ) design
exists is often called the ”spectrum”. The determination of the spectrum for symmetric
designs is a widely open problem. For example, a finite projective plane of order n is a
symmetric design with parameters (n2 + n + 1, n + 1, 1) and it is still unknown whether
finite projective planes of non–prime–power order may exist at all.

The existence/non-existence of a symmetric design has often required ”ad hoc” treat-
ments even for a single parameter set (v, k, λ). The most famous instance of this circum-
stance is perhaps the non-existence of the projective plane of order 10, see [9].

It is of interest to study symmetric designs with additional properties, which often
involve the assumption that a non–trivial automorphism group acts on the design under
consideration, see for instance [3].

Among symmetric block designs of square order, a study of symmetric block designs of
order 49 is of a particular interest. There are 15 possible parameters (v, k, λ) for symmetric
designs of order 49, but until now only a few results are known (see [2], [4]). Due to the
fact that symmetric designs of order 49 have a big number of points (blocks), the study of

DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3769

Email addresses: menderes gashi@yahoo.com (M. Gashi)

https://www.ejpam.com 608 c© 2020 EJPAM All rights reserved.



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 609

sporadic cases is very difficult, except, possibly, when the existence of a collineation group
is assumed.

A few methods for the construction of symmetric designs are known and all of them
have shown to be effective in certain situations. Here, we shall use the method of tactical
decompositions, assuming that a certain automorphism group acts on the design we want
to construct, used by Z. Janko in [7]; see also [3] and [5]. The present paper is concerned
with a symmetric design D = (P,B, I) with parameters (280, 63, 14): the existence/non–
existence of such a design is still in doubt as far as we know. We shall further assume
that the given design admits a certain automorphism group of order 93. We assume the
reader is familiar with the basic facts of design theory, see for instance [8], [1] and [10].
If g is an automorphism of a symmetric design D with parameters (v, k, λ), then g fixes
an equal number of points and blocks, see [10, Theorem 3.1, p.78]. We denote the sets
of these fixed elements by FP(g) and FB(g) respectively, and their cardinality simply by
|F (g)|. We shall make use of the following upper bound for the number of fixed points,
see [10, Corollary 3.7, p. 82]:

|F (g)| ≤ k +
√
k − λ. (1)

It is also known that an automorphism group G of a symmetric design has the same
number of orbits on the set of points P as on the set of blocks B: [10, Theorem 3.3, p.79].
Denote that number by t.

2. Point- and block-orbits

We adopt the notation and terminology of Section 1 in [3]. In the following, for the
sake of completeness, some fundamental relations are explicitly provided. Let D be a
symmetric design with parameters (v, k, λ) and let G be a subgroup of the automorphism
group Aut(D) of D. Denote the point orbits of G on P by P1,P2, . . .Pt and the line orbits
of G on B by B1,B2, . . .Bt . Put |Pr| = ωr and |Bi| = Ωi. Obviously,

t∑
r=1

ωr =
t∑
i=1

Ωi = v. (2)

Let γir be the number of points from Pr, which lie on a line from Bi; clearly this
number does not depend on the chosen line. Similarly, let Γjs be the number of lines from
Bj which pass through a point from Ps. Then, obviously,

t∑
r=1

γir = k and
t∑

j=1

Γjs = k. (3)

By [1, Lemma 5.3.1. p.221], the partition of the point set P and of the block set B forms
a tactical decomposition of the design D in the sense of [1, p.210]. Thus, the following
equations hold:



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 610

Ωi · γir = ωr · Γir (4)

t∑
r=1

γirΓjr = λΩj + δij(k − λ) (5)

t∑
i=1

Γirγis = λωs + δrs(k − λ) (6)

where δij , δrs are the Kronecker symbols.
For a proof of these equations, the reader is referred to [1] and [3]. Equation (5),

together with (4) yields

t∑
r=1

Ωj

ωr
γirγjr = λΩj + δij(k − λ). (7)

Definition 1. We denote

[Li, Lj ] =
t∑

r=1

Ωj

ωr
γirγjr, 1 ≤ i, j ≤ t

and call these expressions the orbit products. The (t × t)-matrix (γir) is called the orbit
structure of the design D.

An automorphism of an orbit structure is a permutation of rows followed by a per-
mutation of columns leaving that matrix unchanged. It is clear that the set of all such
automorphisms is a group, which we call the automorphism group of that orbit structure.

The first step in the construction of a design is to find all possible orbit structures.
The second step of the construction is usually called indexing. In fact for each coefficient
γir of the orbit matrix one has to specify which γir points of the point orbit Pr lie on the
lines of the block orbit Bi. Of course, it is enough to do this for a representative of each
block orbit, as the other lines of that orbit can be obtained by producing all G-images of
the given representative.

3. Action of the Frobenius group of order 57

In our construction of symmetric 2 − (210, 77, 28) designs we assume the existence of
an automorphism group G = 〈ρ, µ|ρ19 = µ3 = 1, ρµ = ρ7〉, which is a so called Frobenius
group of order 57 with Frobenius kernel of order 19 (see [6]).

Theorem 1. Up to isomorphism there are exactly six orbit structures for symmetric
(210,77,28) designs and the automorphism group G = F19·3 acting with the orbit dis-
tribution O = [1; 19; 19; 19; 19; 19; 19; 19; 19; 19; 19; 19].



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 611

Proof. We denote by ∞ the fixed point of ρ and put PI = {I0, I1, · · · , I18}, I =
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, for the non–trivial orbits of the group G. Thus, G acts on these
point orbits as a permutation group in a unique way. Hence, for the two generators of G
we may put

ρ = (∞)(I0, I1, · · · , I18), I = 1, 2, · · · , 11,

where ∞ is the fixed point of collineation, whereas non–trivial 〈ρ〉−orbits are numbers
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and ∞, 10, 11, · · · , 1118 all points of the symmetric block design
D, and the collineation µ of order 3 acts in the symmetric block design as permutation
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) on orbit numbers, whereas on indices acts µ : x → 7x
(mod 19) or

µ = (∞)(K0)(K1,K7,K11)(K2,K14,K3)(K4,K9,K6)(K5,K16,K17)

(K8,K18,K12)(K10,K13,K15), K = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

We immediately obtain the following.

Corollary 1. The element µ of G of order 3 fixes precisely 12 points and 12 blocks of D.
Each block orbit contains a unique block stabilized by µ.

In what follows, we are going to construct a representative block for each block orbit.
A representative block for the block orbit of length 19 will be the block fixed by µ. Hence
the multiplicities of orbit numbers in orbit blocks, will be ≡ 0, 1 (mod 3).

The 〈ρ〉−fixed block can be writen in the form:

L1 =∞(1011 · · · 118)(2021 · · · 218)(3031 · · · 318)(4041 · · · 418)

or

L1 =∞119219319419.

Let L2, L3, L4, L5, L6, L7, L8, L9, L10, L11, L12 be the representative blocks for the eleven
non–trivial block orbits. There are exactly four non–fixed orbit blocks passing through
the fixed point ∞. Let them be L2, L3, L4, L5. We write

L2 =∞1a12a23a34a45a56a67a78a89a910a1011a11

L3 =∞1b12b23b34b45b56b67b78b89b910b1011b11

L4 =∞1c12c23c34c45c56c67c78c89c910c1011c11

L5 =∞1d12d23d34d45d56d67d78d89d910d1011d11

where ai, bi, ci, di denote the multiplicities of the appearance of orbit numbers in the orbit
blocks L2, L3, L4, L5, respectively.

The multiplicities of the appearances of orbit numbers satisfy the following conditions:



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 612

a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9 + a10 + a11 = 76.

b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8 + b9 + b10 + b11 = 76.

c1 + c2 + c3 + c4 + c5 + c6 + c7 + c8 + c9 + c10 + c11 = 76.

d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 + d9 + d10 + d11 = 76.

Because |Li ∩ L1| = 28, i = 2, 3, 4, 5 and ∞ ∈ Li, i = 1, 2, 3, 4, 5 we have a1 + a2 +
a3 + a4 = 27, b1 + b2 + b3 + b4 = 27 , c1 + c2 + c3 + c4 = 27, d1 + d2 + d3 + d4 = 27 and
consequently a5 + a6 + a7 + a8 + a9 + a10 + a11 = 49, b5 + b6 + b7 + b8 + b9 + b10 + b11 = 49,
c5 + c6 + c7 + c8 + c9 + c10 + c11 = 49, d5 + d6 + d7 + d8 + d9 + d10 + d11 = 49. From (7)
we have

[L2, L2] = 19/1 ·1 ·1+19/19 ·a21 +19/19 ·a22 +19/19 ·a23 +19/19 ·a24 +19/19 ·a25 +19/19 ·a26

+19/19 · a27 + 19/19 · a28 + 19/19 · a29 + 19/19 · a210 + 19/19 · a211 = 28 · 19 + 77− 28 = 581

[L3, L3] = 19/1 · 1 · 1 + 19/19 · b21 + 19/19 · b22 + 19/19 · b23 + 19/19 · b24 + 19/19 · b25 + 19/19 · b26
+19/19 · b27 + 19/19 · b28 + 19/19 · b29 + 19/19 · b210 + 19/19 · b211 = 28 · 19 + 77− 28 = 581

[L4, L4] = 19/1 · 1 · 1 + 19/19 · c21 + 19/19 · c22 + 19/19 · c23 + 19/19 · c24 + 19/19 · c25 + 19/19 · c26
+19/19 · c27 + 19/19 · c28 + 19/19 · c29 + 19/19 · c210 + 19/19 · c211 = 28 · 19 + 77− 28 = 581

[L5, L5] = 19/1 ·1 ·1+19/19 ·d21 +19/19 ·d22 +19/19 ·d23 +19/19 ·d24 +19/19 ·d25 +19/19 ·d26
+19/19 · d27 + 19/19 · d28 + 19/19 · d29 + 19/19 · d210 + 19/19 · d211 = 28 · 19 + 77− 28 = 581

[L2, L3] = 19 · 1 · 1 + 19/19 · a1b1 + 19/19 · a2b2 + 19/19 · a3b3 + 19/19 · a4b4 + 19/19 · a5b5
+19/19·a6b6+19/19·a7b7+19/19·a8b8+19/19·a9b9+19/19·a10b10+19/19·a11b11 = 28·19 = 532

[L2, L4] = 19 · 1 · 1 + 19/19 · a1c1 + 19/19 · a2c2 + 19/19 · a3c3 + 19/19 · a4c4 + 19/19 · a5c5
+19/19·a6c6+19/19·a7c7+19/19·a8c8+19/19·a9c9+19/19·a10c10+19/19·a11c11 = 28·19 = 532

[L2, L5] = 19 · 1 · 1 + 19/19 · a1d1 + 19/19 · a2d2 + 19/19 · a3d3 + 19/19 · a4d4 + 19/19 · a5d5
+19/19·a6d6+19/19·a7d7+19/19·a8d8+19/19·a9d9+19/19·a10d10+19/19·a11d11 = 28·19 = 532

where 0 ≤ ai ≤ 23, i = 1, 2, · · · , 11.
In order to reduce isomorphic cases that may appear in the orbit structures at the last

stage, without loss of generality, for the block L2, we can use the reduction a1 ≥ a2 ≥
a3 ≥ a4, a5 ≥ a6 ≥ a7 ≥ a8 ≥ a9 ≥ a10 ≥ a11.

Using the computer we have proved that there exist exactly twenty one different orbit
types for the block L2 satisfying the above mentioned conditions:



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 613

a1 a2 a3 a4 a5 a6 a7 a8 a9 a19 a11
1. 10 9 4 4 9 7 7 7 7 6 6
2. 10 7 7 3 10 7 7 7 6 6 6
3. 10 7 7 3 9 9 7 6 6 6 6
4. 10 7 6 4 10 9 6 6 6 6 6
5. 10 7 6 4 10 7 7 7 7 7 4
6. 10 7 6 4 9 9 7 7 7 6 4
7. 9 9 6 3 10 7 7 7 6 6 6
8. 9 9 6 3 9 9 7 6 6 6 6
9. 9 7 7 4 10 9 7 7 6 6 4

10. 9 7 7 4 9 9 9 6 6 6 4
11. 9 7 7 4 9 9 7 7 7 7 3
12. 9 6 6 6 12 7 6 6 6 6 6
13. 9 6 6 6 10 10 7 6 6 6 4
14. 9 6 6 6 10 9 7 7 7 6 3
15. 9 6 6 6 9 9 9 7 7 4 4
16. 9 6 6 6 9 9 9 7 6 6 3
17. 7 7 7 6 12 7 7 7 6 6 4
18. 7 7 7 6 10 10 7 7 7 4 4
19. 7 7 7 6 10 10 7 7 6 6 3
20. 7 7 7 6 10 9 9 7 6 4 4
21. 7 7 7 6 10 9 9 6 6 6 3

Further on, we find the possible candidates for L3 considering each orbit type for L2.
Among the candidates for block L3 are blocks L4, L5. Therefore, for each case for

L2, from the candidates for L3, must be found triples of blocks {L3, L4, L5}, which are
pairwise compatibile.

Using the computer we obtain the results which we present in the table below:



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 614

Table 1:

Block L2
Number of orbit
types for L3

Number of quadru-
ples {L2, L3, L4, L5}

Type 1. 969 8

Type 2. 1201 15

Type 3. 2073 41

Type 4. 1847 13

Type 5. 2355 6

Type 6. 1172 57

Type 7. 1401 8

Type 8. 1792 21

Type 9. 1001 67

Type 10. 2001 43

Type 11. 1301 23

Type 12. 1170 0

Type 13. 1371 11

Type 14. 1295 35

Type 15. 1584 8

Type 16. 1902 55

Type 17. 1001 11

Type 18. 2340 13

Type 19. 1026 28

Type 20. 1131 46

Type 21. 2098 69

Therefore, there are 578 quadruples L2, L3, L4, L5 (i.e. 578 quintuples L1, L2, L3, L4, L5).

The sixth orbit block L6 has the form:

L6 = 1e12e23e34e45e56e67e78e89e910e1011e11

where ei, i = 1, 2, · · · , 11 are multiplicities of the appearance of orbit numbers 1,2,3,
4,5,6,7,8,9,10 and 11 in orbit block L6.

We have: e1 + e2 + e3 + e4 + e5 + e6 + e7 + e8 + e9 + e10 + e11 = 77,

[L6, L6] = e21 + e22 + e23 + e24 + e25 + e26 + e27 + e28 + e29 + e210 + e211 = 28 · 19 + 77− 28 = 581,

[L6, Li] = 28 · 19 = 532, (i = 2, 3, 4, 5).

[L6 ∩L1] = 28 implies e1 + e2 + e3 + e4 = 28, therefore c5 + c6 + c7 + c8 + c9 + c10 + c11 =
77− 28 = 49.

[L6, L6] = 581 implies 0 ≤ ci ≤ 24, i = 1, 2, · · · , 11.



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 615

Candidates for L6 are also blocks L7, L8, L9, L10, L11, and L12, so we look for septuples
{L6, L7, L8, L9, L10, L11, L12} which are pairwise compatibile. Using the computer for the
number of quadruples {L2, L3, L4, L5} given in Table 1, we obtain the results which we
present in the table below:

Type
for L2

Subcase for
quadruples
{L2, L3, L4, L5}

Number of or-
bit types for
L6

Number of septuples
{L6, L7, L8, L9, L10, L11, L12}

1 1 48 0
1 2 48 0
2 3 40 0
2 4 40 0
2 6 60 0
2 8 60 0
3 3 32 0
3 4 22 0
3 12 78 0
3 13 33 0
3 14 20 0
3 15 23 0
3 16 23 0
3 18 33 0
3 19 79 0
5 1 120 12
5 6 48 0
6 42 48 0
7 8 25 0
9 22 22 0
9 59 7 1
9 60 22 1
9 63 36 0
10 41 45 4
11 1 48 0
11 2 48 0
11 10 32 0
11 17 39 0
13 7 45 4
13 10 32 0
13 11 39 0
14 4 25 0
15 3 22 1
15 5 25 0
15 6 25 0
15 7 36 0
15 8 36 0
16 23 36 0
16 24 7 1
16 29 39 0
16 54 74 0
16 55 74 0



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 616

Type
for L2

Subcase for
quadruples
{L2, L3, L4, L5}

Number of or-
bit types for L6

Number of septuples
{L6, L7, L8, L9, L10, L11, L12}

17 3 60 0
17 5 23 0
17 11 36 0
18 1 40 0
18 2 78 0
18 3 60 0
19 5 33 0
19 6 20 0
19 15 22 1
19 20 32 0
19 21 36 0
19 22 22 1
19 23 32 0
19 26 74 0
19 27 36 0
20 7 40 0
20 9 33 0
20 10 23 0
20 34 32 0
21 11 78 0
21 43 36 0
21 45 7 1
21 46 39 0
21 47 45 4
21 48 7 1
21 56 39 0
21 57 7 1
21 59 39 0
21 61 7 1
21 63 74 0
21 64 39 0
21 65 45 4
21 66 36 0

Other subcases < 7 0

From the table above it can be seen that there are 75 subcases, for which the number of
orbit types for L6 is greater than 6, and for those subcases we have searched for septuples
{L6, L7, L8, L9, L10, L11, L12}, respectively, we have searched for orbit structures. For these
75 subcases we have found 38 orbit structures. After the removal of the isomorphic cases,
there remained exactly 6 orbit structures, which are given below:



M. Gashi / Eur. J. Pure Appl. Math, 13 (3) (2020), 608-619 617

Table 2:

OS1. 1 19 19 19 19 19 19 19 19 19 19 19

1 19 19 19 19 0 0 0 0 0 0 0
1 10 7 6 4 10 7 7 7 7 7 4
1 7 10 4 6 4 7 7 7 7 7 10
1 6 4 7 10 7 10 7 7 7 4 7
1 4 6 10 7 7 4 7 7 7 10 7
0 10 4 7 7 7 4 9 7 6 6 10
0 7 7 10 4 4 10 9 7 6 7 6
0 7 7 7 7 9 9 3 6 4 9 9
0 7 7 7 7 7 7 6 3 12 7 7
0 7 7 7 7 6 6 4 12 9 6 6
0 7 7 4 10 6 7 9 7 6 10 4
0 4 10 7 7 10 6 9 7 6 4 7

OS2. 1 19 19 19 19 19 19 19 19 19 19 19

1 19 19 19 19 0 0 0 0 0 0 0
1 10 7 6 4 10 7 7 7 7 7 4
1 7 10 4 6 4 7 7 7 7 7 10
1 6 4 7 10 7 10 7 7 7 4 7
1 4 6 10 7 7 4 7 7 7 10 7
0 10 4 7 7 6 4 10 7 7 6 9
0 7 7 10 4 4 9 6 9 9 6 6
0 7 7 7 7 9 7 4 10 3 7 9
0 7 7 7 7 9 7 4 3 10 7 9
0 7 7 7 7 6 10 9 4 4 10 6
0 7 7 4 10 6 6 6 9 9 9 4
0 4 10 7 7 9 6 10 7 7 4 6

OS3. 1 19 19 19 19 19 19 19 19 19 19 19

1 19 19 19 19 0 0 0 0 0 0 0
1 7 7 7 6 10 9 9 6 6 6 3
1 7 7 7 6 3 9 9 6 6 6 10
1 7 7 4 9 9 6 4 9 6 6 9
1 6 6 9 6 6 4 6 7 10 10 6
0 10 10 4 4 7 6 7 6 9 7 7
0 10 3 6 9 6 9 6 6 7 9 6
0 7 7 10 4 7 9 4 10 6 6 7
0 7 7 7 7 7 4 10 9 3 9 7
0 6 6 9 7 10 6 7 3 7 6 10
0 6 6 7 9 6 6 9 9 10 3 6
0 3 10 6 9 6 9 6 6 7 9 6

OS4. 1 19 19 19 19 19 19 19 19 19 19 19

1 19 19 19 19 0 0 0 0 0 0 0
1 7 7 7 6 10 9 9 6 6 6 3
1 7 7 7 6 3 9 9 6 6 6 10
1 7 7 4 9 9 6 4 9 6 6 9
1 6 6 9 6 6 4 6 7 10 10 6
0 10 9 6 3 6 7 6 10 7 7 6
0 9 6 4 9 6 9 6 4 9 9 6
0 9 3 10 6 9 7 6 6 6 6 9
0 6 10 6 6 9 4 9 4 7 7 9
0 6 6 7 9 6 6 9 9 10 3 6
0 6 6 7 9 6 6 9 9 3 10 6
0 3 9 9 7 7 10 4 7 7 7 7



REFERENCES 618

OS5. 1 19 19 19 19 19 19 19 19 19 19 19

1 19 19 19 19 0 0 0 0 0 0 0
1 9 7 7 4 10 9 7 7 6 6 4
1 6 7 7 7 9 3 6 6 10 6 9
1 6 7 7 7 3 9 6 6 9 10 6
1 6 6 6 9 6 7 9 9 3 6 9
0 10 6 6 6 6 7 10 3 7 7 9
0 10 6 6 6 6 7 3 10 7 7 9
0 7 6 6 9 6 7 9 9 10 4 4
0 6 10 3 9 9 7 6 6 6 9 6
0 6 9 9 4 6 4 9 9 6 9 6
0 6 3 10 9 9 7 6 6 6 9 6
0 4 9 9 6 7 10 6 6 7 4 9

OS6. 1 19 19 19 19 19 19 19 19 19 19 19

1 19 19 19 19 0 0 0 0 0 0 0
1 9 7 7 4 10 9 7 7 6 6 4
1 6 7 7 7 7 6 7 6 10 3 10
1 6 7 7 7 7 6 7 6 3 10 10
1 6 6 6 9 4 7 7 9 9 9 4
0 10 6 6 6 6 7 3 10 7 7 9
0 10 6 6 6 6 6 9 3 9 9 7
0 7 6 6 9 10 3 9 9 6 6 6
0 6 10 3 9 7 10 7 6 6 6 7
0 6 9 9 4 4 7 10 9 6 6 7
0 6 3 10 9 7 10 7 6 6 6 7
0 4 9 9 6 9 6 4 6 9 9 6

�

Remark 1. The actual indexing of these six orbit structures in order to produce an ex-
ample is still an open problem.

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