403 IHJPAS. 37 (1) 2024 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq PISSN: 1609-4042, EISSN: 2521-3407 1Mohammed Salah Aldeen Zidan* 2Niran Sabah Jasim 3Azza I.M.S. Abu-Shams 4Ahmad Issa 1 Ministry of Education, Directorate General of Education karkh 1, Baghdad, Iraq. 2 University of Baghdad, College of Education for Pure Science Ibn Al-Haitham, Department of Mathematics, Baghdad, Iraq. 3 Philadelphia University, College of Science, Mathematics Department, Ammaan Jordan. 4 Karabรผk University, Faculty of Science, Department of Mathematics, Karabรผk, Tรผrkiye. *Corresponding Author. niraan.s.j@ihcoedu.uobaghdad.edu.iq Abstract The set of all (nร—n) non-singular matrices over the field F this set forms a group under the operation of matrix multiplication. This group is called the general linear group of dimension n over the field F, denoted by GL(n, F). The determinant of these matrices is a homomorphism from GL(n, F) into F* and the kernel of this homomorphism was the special linear group and denoted by ๐’ฎโ„’(n, F). Thus ๐’ฎโ„’(n, F) is the subgroup of GL(n, F) which contains all matrices of determinant one. The rational valued characters of the rational representations written as a linear combination of the induced characters for the groups ๐’ฎโ„’(2, 57) discuss in this paper and find the Artin indicator for this group after study the rational valued characters of the rational representations and the induced characters. Keywords: Artin indicator, induced characters table, rational character table. 1. Introduction The group of all matrices of determinant 1 is symbolize by ๐’ฎโ„’(n,F), [1,2], searchers in [3] define the representation of the group. Authors in [4,5] study the character table of rational representations for the group ๐’ฎโ„’(2, ๐‘), while the authors in [6] study and find the periodical split for the groups PSL(2,31) and PSL(2,37) and the authors in [7] survey and get the calculation for the groups SL(2,U), U = 31 and 37. While the authors in [8] obtain the same results for the ๐’ฎ๐’ฐ๐’ฏ(2, ๐“…), where ๐“… = 3, 5, 7. Searchers in [10,11] study the New algorithm for number recognition and Symmetric generalized respectively. We apply the same idea in [4-8] to compute the character table of rational representations for the group ๐’ฎโ„’(2, 57). Also we apply the same idea in [9-13] to compute the Received 20/September/2022, Received 21/February/2023, Accepted 28/February/2023, Published 20/January/2024 Outcome for The Group SL(2,57) doi.org/10.30526/37.1.3018 https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042 https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407 mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0002-2816-304X mailto:mohammed.salahaldeen1203a@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0001-5340-3020 mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq https://orcid.org/0009-0003-1741-1447 mailto:aabushams@philadelphia.edu.jo https://orcid.org/0000-0001-7495-3443 mailto:ahmad93.issa18@gmail.com IHJPAS. 37 (1) 2024 404 Artin indicator for this group. We count all cyclic subgroup, Artin indicator, the rational valued characters of the rational representations and the induced characters for the group ๐’ฎโ„’(2, 57) in this work. 2. Basic Concepts Theorem 2.1: [14-16] ๏‚ฝ ๐’ฎโ„’(2, ๐“†๐“ƒ) ๏‚ฝ= ๐“†๐“ƒ ( ๐“†2๐“ƒ โ€“ 1). Definition 2.2: [17-21] Let H be a cyclic subgroup of a group G, and ๏ฆ be a class function of H. Then m GG i i 1H C (g) (g) (x ) C (g) = ๏ฆ ๏‚ญ = ๏ฆ๏ƒฅ Definition 2.3: [22-25] The character induced from the principal character of a cyclic subgroups of G is called Artin character. Definition 2.4: [26-30] Let G be a finite group and let ๏ฃ be any rational valued character on G. The smallest positive number n such that, c c c n a๏ฃ = ๏ฆ๏ƒฅ , where ac๏ƒŽZ and ๏ฆc is Artin character, is called the Artin exponent of G and denoted by A(G). 3- The Outcome Authors in [4-8] study the character table of rational representations for the group ๐’ฎโ„’(2, ๐‘) we apply that idea and compute the character table of rational representations for the group ๐’ฎโ„’(2, 57). Also we apply the same idea in [4,9] to compute the Artin indicator for this group. The character table of rational representations for the group ๐’ฎโ„’(2, 57) is Table 1. The character table of rational representations for the group ๐’ฎโ„’(2, 57) Cg 1 z c = d zc = zd a a 2 a 4 a 19531 |Cg | 1 1 3051757812 3051757812 6103593750 6103593750 6103593750 6103593750 |CG(g) | 47683715812500 0 4768371581250 00 156250 156250 78124 78124 78124 78124 1G 1 1 1 1 1 1 1 1 ฯˆ 78125 78125 0 0 1 1 1 1 ฯ‡1 3051601560 -3051601560 39060 -39060 0 4 -4 0 ฯ‡2 762900390 -762900390 9765 9765 1 -1 0 -19530 ฯ‡4 762900390 762900390 9765 9765 -1 0 0 19530 ฯ‡19531 156252 -156252 2 -2 0 -4 4 4 1ฮธ 1959974912 -1959974912 -25088 25088 0 0 0 0 2ฮธ 979987456 979987456 -12544 -12544 0 0 0 0 3ฮธ 979987456 -979987456 -12544 12544 0 0 0 0 6ฮธ 489993728 489993728 -6272 -6272 0 0 0 0 29ฮธ 69999104 -69999104 -896 896 0 0 0 0 58ฮธ 34999552 34999552 -448 -448 0 0 0 0 ฮธ87 34999552 -34999552 -448 448 0 0 0 0 ฮธ174 17499776 17499776 -224 -224 0 0 0 0 ฮธ449 4374944 -4374944 -56 56 0 0 0 0 ฮธ898 2187472 2187472 -28 -28 0 0 0 0 ฮธ1347 2187472 -2187472 -28 28 0 0 0 0 ฮธ2694 1093736 1093736 -14 -14 0 0 0 0 ฮธ13021 156248 -156248 -2 2 0 0 0 0 IHJPAS. 37 (1) 2024 405 Complete Table 1. The character table of rational representations for the group ๐’ฎโ„’(2, 57) Complete Table 1. The character table of rational representations for the group ๐’ฎโ„’(2, 57) ฮธ26042 78124 78124 -1 -1 0 0 0 0 ฮพ 78126 78126 1 1 -2 2 2 -2 ฮท 156248 -156248 -1 1 0 0 0 0 Cg b b 2 b 3 b 6 b 29 58b 87b |Cg | 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 |CG(g)| 78126 78126 78126 78126 78126 78126 78126 1G 1 1 1 1 1 1 1 ฯˆ -1 -1 -1 -1 -1 -1 -1 ฯ‡1 0 0 0 0 0 0 0 ฯ‡2 0 0 0 0 0 0 0 ฯ‡3 0 0 0 0 0 0 0 ฯ‡4 0 0 0 0 0 0 0 1ฮธ -2 2 4 -4 56 -56 -112 2ฮธ 1 0 -2 0 -28 0 56 3ฮธ 2 2 0 0 -56 56 0 4ฮธ -1 0 0 0 28 0 0 5ฮธ 2 -2 -4 4 0 0 224 6ฮธ -1 0 2 0 0 0 -112 ฮธ7 -2 2 0 0 112 -112 -224 ฮธ8 1 0 0 0 0 0 112 ฮธ 2 -2 -4 4 -56 56 112 ฮธ -1 0 2 -7 28 56 -56 ฮธ -2 2 0 -14 56 -56 -56 ฮธ 1 0 -7 7 -28 -28 -28 ฮธ -2 2 4 -4 -4 -4 -4 ฮธ 1 2 -2 -2 -2 -2 -2 ฮพ 0 0 0 0 0 0 0 ฮท 4 -4 4 -4 4 -4 4 Cg b174 b 449 b 898 b 1347 b 2694 13021b 26042b |Cg | 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 |CG(g)| 78126 78126 78126 78126 78126 78126 78126 1G 1 1 1 1 1 1 1 ฯˆ -1 -1 -1 -1 -1 -1 -1 ฯ‡1 0 0 0 0 0 0 0 ฯ‡2 0 0 0 0 0 0 0 ฯ‡3 0 0 0 0 0 0 0 ฯ‡4 0 0 0 0 0 0 0 1ฮธ 112 896 -896 -1792 1792 -25088 25088 2ฮธ 0 -448 0 896 -3136 12544 25088 3ฮธ 0 -896 896 0 -6272 25088 -25088 4ฮธ 0 448 -1568 -3136 3136 -12544 -12544 5ฮธ 0 -896 896 1792 -1792 -1792 -1792 6ฮธ 0 448 896 -896 -896 -896 -896 ฮธ7 224 896 -896 -896 -896 -896 -896 ฮธ8 224 -448 -448 -448 -448 -448 -448 ฮธ -112 -112 -112 -112 -112 -112 -112 ฮธ -56 -56 -56 -56 -56 -56 -56 ฮธ -56 -56 -56 -56 -56 -56 -56 ฮธ -28 -28 -28 -28 -28 -28 -28 ฮธ -4 -4 -4 -4 -4 -4 -4 ฮธ -2 -2 -2 -2 -2 -2 -2 ฮพ 0 0 0 0 0 0 0 ฮท -4 4 -4 4 -4 4 -4 IHJPAS. 37 (1) 2024 406 This group has 22 cyclic subgroups generated by the conjugacy classes of the group. The induced character table for this group is: Table 2. The induced character table for the group ๐’ฎโ„’(2, 57) Cg 1 z c = d zc = zd a a 2 a 4 a 19531 |Cg | 1 1 3051757812 3051757812 6103593750 6103593750 6103593750 6103593750 |CG(g)| 476837158125000 47683715812500 0 156250 156250 78124 78124 78124 78124 ๏†1 476837158125000 0 0 0 0 0 0 0 ๏†2 95367431625000 95367431625000 0 0 0 0 0 0 ๏†3 6103515624 0 2 0 0 0 0 0 ๏†4 6103515624 3051757812 3 3 0 0 0 0 ๏†5 6103593750 12207187500 0 0 2 0 0 0 ๏†6 12207187500 24414375000 0 0 0 4 0 0 7๏† 24414375000 0 0 0 0 0 8 0 8๏† 119209289531250 23841857906250 0 0 0 0 0 0 39062 9๏† 12206875000 24413750000 0 0 0 0 0 0 10๏† 24413750000 0 0 0 0 0 0 0 11๏† 36620625000 73241250000 0 0 0 0 0 0 12๏† 73241250000 0 0 0 0 0 0 0 ๏†13 353999375000 707998750000 0 0 0 0 0 0 ๏†14 707998750000 0 0 0 0 0 0 0 ๏†15 1061998125000 2123996250000 0 0 0 0 0 0 ๏†16 2123996250000 0 0 0 0 0 0 0 ๏†17 5480886875000 10961773750000 0 0 0 0 0 0 ๏†18 10961773750000 0 0 0 0 0 0 0 ๏†19 16442660625000 32885321250000 0 0 0 0 0 0 ๏†20 32885321250000 0 0 0 0 0 0 0 ๏†21 158945719375000 31789143875000 0 0 0 0 0 0 0 ๏†22 317891438750000 0 0 0 0 0 0 0 IHJPAS. 37 (1) 2024 407 Complete Table 2. The induced character table for the group ๐’ฎโ„’(2, 57) Cg b b 2 b 3 b 6 b 29 58b 87b |Cg | 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 |CG(g)| 78126 78126 78126 78126 78126 78126 78126 ๏†1 0 0 0 0 0 0 0 ๏†2 0 0 0 0 0 0 0 ๏†3 0 0 0 0 0 0 0 ๏†4 0 0 0 0 0 0 0 ๏†5 0 0 0 0 0 0 0 ๏†6 0 0 0 0 0 0 0 7๏† 0 0 0 0 0 0 0 8๏† 0 0 0 0 0 0 0 9๏† 2 0 0 0 0 0 0 10๏† 0 2 0 0 0 0 0 11๏† 0 0 6 0 0 0 0 12๏† 0 0 0 12 0 0 0 ๏†13 0 0 0 0 58 0 0 ๏†14 0 0 0 0 0 116 0 ๏†15 0 0 0 0 0 0 174 ๏†16 0 0 0 0 0 0 0 ๏†17 0 0 0 0 0 0 0 ๏†18 0 0 0 0 0 0 0 ๏†19 0 0 0 0 0 0 0 ๏†20 0 0 0 0 0 0 0 ๏†21 0 0 0 0 0 0 0 ๏†22 0 0 0 0 0 0 0 IHJPAS. 37 (1) 2024 408 Complete Table 2. The induced character table for the group ๐’ฎโ„’(2, 57) Hence, the rational valued characters in the first tables is written as a linear combination of induced characters in the second table 1 = 1 5208 ๏†22 + 1 26042 ๏†21 + 1 5388 ๏†20 + 1 2694 ๏†19 + 1 1769 ๏†18 + 1 898 ๏†17 + 1 348 ๏†16 + 1 174 ๏†15 + 1 116 ๏†14 + 1 58 ๏†13+ 1 12 ๏†12+ 1 6 ๏†11+ 1 2 ๏†10 + 1 2 ๏†9 + 1 39062 ๏†8 + 1 8 ๏†7 + 1 4 ๏†6+ 1 2 ๏†5 + 1 3 ๏†4 โˆ’7.4545390170038๏†2โˆ’2.5350887732113๏†1, ฮจ=โˆ’ 1 5208 ๏†22โˆ’ 1 26042 ๏†21โˆ’ 1 5388 ๏†20โˆ’ 1 2694 ๏†19โˆ’ 1 1796 ๏†18โˆ’ 1 898 ๏†17โˆ’ 1 348 ๏†16โˆ’ 1 174 ๏†15โˆ’ 1 116 ๏†14โˆ’ 1 58 ๏†1 3- 1 12 ๏†12โˆ’ 1 6 ๏†11โˆ’ 1 2 ๏†10โˆ’ 1 2 ๏†9 + 1 39062 ๏†8 + 1 8 ๏†7 + 1 4 ๏†6 + 1 2 ๏†5 + 8.1974976391737๏†2 + 3.2782431357639๏†1, ๏ฃ1 = โˆ’ 1 2 ๏†7 + ๏†6โˆ’13020๏†4 + 39060 ๏†3+ 39706420735680 95376431625000 ๏†2 + 238031060800780 476837158125000 ๏†1, ๏ฃ2 = โˆ’0.49997 ๏†8 โˆ’ 1 4 ๏†6 + 1 2 ๏†5 + 3255๏†4 + 109269428196208 95376431625000 ๏†2+ 149031782399292 476837158125000 ๏†1, ๏ฃ4 = 0.49997๏†8โˆ’ 1 2 ๏†5 + 3255 ๏†4 + 189275531789958 95376431625000 ๏†2 + 109811334644164 476837158125000 ๏†1, Cg b 174 b 449 b 898 b 1347 b 2694 13021b 26042b |Cg | 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 |CG(g)| 78126 78126 78126 78126 78126 78126 78126 ๏†1 0 0 0 0 0 0 0 ๏†2 0 0 0 0 0 0 0 ๏†3 0 0 0 0 0 0 0 ๏†4 0 0 0 0 0 0 0 ๏†5 0 0 0 0 0 0 0 ๏†6 0 0 0 0 0 0 0 7๏† 0 0 0 0 0 0 0 8๏† 0 0 0 0 0 0 0 9๏† 0 0 0 0 0 0 0 10๏† 0 0 0 0 0 0 0 11๏† 0 0 0 0 0 0 0 12๏† 0 0 0 0 0 0 0 ๏†13 0 0 0 0 0 0 0 ๏†14 0 0 0 0 0 0 0 ๏†15 0 0 0 0 0 0 0 ๏†16 348 0 0 0 0 0 0 ๏†17 0 898 0 0 0 0 0 ๏†18 0 0 1796 0 0 0 0 ๏†19 0 0 0 2694 0 0 0 ๏†20 0 0 0 0 5388 0 0 ๏†21 0 0 0 0 0 26042 0 ๏†22 0 0 0 0 0 0 5208 IHJPAS. 37 (1) 2024 409 ๏ฃ19531 = 0.00001๏†8 + 1 2 ๏†7 โˆ’ ๏†6-0.66667๏†4 + 2๏†3 + 0.0002523350784 ๏†2 + 0.0002613036685 ๏†1, ๏ฑ1 = 4.81720๏†22 โˆ’ 0.96337๏†21 + 0.33259๏†20 โˆ’ 0.66518๏†19 โˆ’0.49889๏†18+ 0.99777๏†17 + 0.32184๏†16 โˆ’ 0.64368๏†15 โˆ’ 0.48276๏†14 + 0.96552๏†13 โˆ’ 1 3 ๏†12+0.66667๏†11 + ๏†10 โˆ’ ๏†9 + 8362.66667๏†4 โˆ’ 25088๏†3 + 3.0562055844106 ๏†2 โˆ’ 2.11532393244476๏†1, ๏ฑ2 = 4.81720๏†22 + 0.48168๏†21 โˆ’ 0.58203๏†20 + 0.33259๏†19 โˆ’ 0.49889๏†17+ 0.32184๏†15 โˆ’ 0.48276๏†13 โˆ’ 1 3 ๏†11 + 1 2 ๏†9 โˆ’ 0.00024๏†4 โˆ’ 1.6663882111459 ๏†2 โˆ’ 3.6712421001151๏†1, ๏ฑ3 = โˆ’ 4.81720๏†22 + 0.96337๏†21-1.16407๏†20+0.49889๏†18-0.99777๏†17+ 0.48276๏†14 โˆ’ 0.96552๏†13 + ๏†10 + ๏†9 + 0.00024๏†4 โˆ’12544๏†3 โˆ’ 294651553870827 95376431625000 ๏†2 + 2.5131832655351๏†1, ๏ฑ6 = โˆ’ 2.40860๏†22 โˆ’ 0.48168๏†21 + 0.58203๏†20 โˆ’1.16407๏†19 โˆ’ 0.87305๏†18+ 0.49889๏†17 + 0.48276๏†13 โˆ’ 1 2 ๏†9 โˆ’ 2090.66667๏†4 + 1919847155587088 95376431625000 ๏†2 + 2016598610418427 476837158125000 ๏†1, ๏ฑ29 = โˆ’ 0.34409๏†22 โˆ’ 0.06881๏†21 โˆ’ 0.33259๏†20 + 0.66518๏†19 + 0.49889๏†18 โˆ’ 0.99777๏†17 + 1.28736๏†15 + 1 3 ๏†12 โˆ’ 0.66667๏†11 โˆ’ ๏†10 + ๏†9 + 298.66667 ๏†4 โˆ’896๏†3 + 7315381744325 95376431625000 ๏†2 + 0.2214571432381๏†1, ๏ฑ58 = โˆ’0.17204๏†22 โˆ’ 0.34406๏†21 โˆ’ 0.16629๏†20 โˆ’ 0.33259๏†19 + 0.49889๏†18 + 0.49889๏†17 โˆ’ 0.64368๏†15 + 1 3 ๏†11โˆ’ 1 2 ๏†9โˆ’149.33334๏†4 + 1.2231960883785 ๏†2 + 0.4830853368404๏†1, ๏ฑ87 = โˆ’ 0.17204๏†22 โˆ’ 0.34406๏†21 โˆ’ 0.33259๏†19 โˆ’ 0.49889๏†18+ 0.99777๏†17 + 0.64368๏†16 โˆ’ 1.28736๏†15 โˆ’ 0.96552๏†14 + 1.93103๏†13 + ๏†10 โˆ’ ๏†9 + 149.33334๏†4 โˆ’ 448๏†3 + 1.156679561213๏†2 + 0.2558004465064๏†1, ๏ฑ174 = โˆ’ 0.08602๏†22 โˆ’ 0.01720๏†21 โˆ’ 0.08315๏†20 โˆ’ 0.16639๏†19 โˆ’ 0.24944๏†18 โˆ’ 0.49889๏†17 + 0.64368๏†16 + 0.64368๏†15 + 1 2 ๏†9 โˆ’ 74.66667๏†4 + 15256741967469 9537641625000 ๏†2 + 54673491989306 476387158125000 ๏†1, ๏ฑ449 = โˆ’ 0.21505๏†22 โˆ’ 0.00430๏†21 โˆ’ 0.02079๏†20 โˆ’ 0.04157๏†19 โˆ’ 0.06236๏†18 โˆ’ 0.12472๏†17 โˆ’ 0.32184๏†16 + 0.64368๏†15 + 2.07143๏†14 โˆ’ 0.96552๏†13 + 1 3 ๏†12 โˆ’ 0.66667๏†11 โˆ’ ๏†10 + ๏†9 + 18.66667๏†4 โˆ’ 56๏†3 + 3884984923184 95376431625000 ๏†2 + 80933664633553 476387158125000 ๏†1, ๏ฑ898 = โˆ’ 1 93 ๏†22 โˆ’ 0.00215๏†21 โˆ’ 0.01039๏†20 โˆ’ 0.02079๏†19 โˆ’ 0.03118๏†18 โˆ’ 0.06236๏†17 โˆ’ 0.16092๏†16 โˆ’ 0.32184๏†15 + 0.48276๏†14 โˆ’ 1 2 ๏†13 โˆ’ 0.58333๏†12 + 1 3 ๏†11 โˆ’ 1 2 ๏†9 โˆ’ 9.33333๏†4 + 9514432370548 45376431625000 ๏†2 + 15253832651739 476387158125000 ๏†1, ๏ฑ1347 = โˆ’ 1 93 ๏†22 โˆ’ 0.00215๏†21 โˆ’ 0.01039๏†20 โˆ’ 0.02079๏†19 โˆ’ 0.03118๏†1 โˆ’ 0.06236๏†17- โˆ’ 0.16092๏†16 โˆ’ 0.32184๏†15 โˆ’ 0.48276๏†14 + 0.96552๏†13 โˆ’ 1.16667๏†12 + ๏†10 โˆ’ ๏†9 + 9.33333๏†4 โˆ’ 28๏†3+ 818622717026 9537641625000 ๏†2 + 6596680087056 476387158125000 ๏†1, IHJPAS. 37 (1) 2024 410 ๏ฑ2694 = โˆ’ 1 186 ๏†22 โˆ’ 0.00108๏†21 โˆ’ 0.00529๏†20 โˆ’ 0.01039๏†19 โˆ’ 0.01559๏†18 โˆ’ 0.03118๏†17 โˆ’ 0.08046๏†16 โˆ’ 0.16092๏†15 โˆ’ 0.24138๏†14โˆ’ 0.48276๏†13 + 0.58333 ๏†12 โˆ’1.16667 ๏†11 + 1 2 ๏†9 โˆ’ 4.66667 ๏†4 + 1797860434764 95376431625000 ๏†2 + 5071174336039 476387158125000 ๏†1, ๏ฑ13021 = โˆ’ 1 1302 ๏†22 โˆ’ 0.00015๏†21- 1 1347 ๏†20 โˆ’ 0.00148๏†19 โˆ’ 1 449 ๏†18 โˆ’ 0.00445๏†17 โˆ’ 1 87 ๏†16 โˆ’ 0.02299๏†15 โˆ’ 1 29 ๏†14 โˆ’ 0.06897๏†13 โˆ’ 1 3 ๏†12 + 0.66667๏†11 + ๏†10 โˆ’ ๏†9 +0.66667๏†4โˆ’ 2๏†3 +0.0002294029092๏†2+0.0014695113016 ๏†1, ๏ฑ26042 = โˆ’ 1 2604 ๏†22 โˆ’ 1 13021 ๏†21โˆ’ 1 2694 ๏†20โˆ’ 1 1347 ๏†19โˆ’ 1 898 ๏†18โˆ’ 1 449 ๏†17โˆ’ 1 174 ๏†16โˆ’ 1 87 ๏†15 โˆ’ 1 58 ๏†14โˆ’ 1 29 ๏†13โˆ’ 1 6 ๏†12โˆ’ 1 3 ๏†11+ ๏†10 + 1 2 ๏†9โˆ’ 1 3 ๏†4 + 135292955728 95376431625000 ๏†2 + 0.0007616100819๏†1, ๏บ = โˆ’ 1 19531 ๏†8 + 1 4 ๏†7 + 1 2 ๏†6โˆ’ 1 2 ๏†5 + 1 3 ๏†4 + 5086419272 95376431625000 ๏†2 + 54932148435 476387158125000 ๏†1, ๏จ = โˆ’ 1 1302 ๏†22 + 0.00015๏†21โˆ’ 1 1347 ๏†20 + 0.00149๏†19 โˆ’ 1 449 ๏†18 + 0.00445๏†17 โˆ’ 1 87 ๏†16 + 0.02299 ๏†15 โˆ’ 1 29 ๏†14+ 0.06897๏†13โˆ’ 1 3 ๏†12+ 0.66667๏†11โˆ’2๏†10 + 2๏†9 + 1 3 ๏†4 โˆ’ ๏†3-0.0075290764184 ๏†2โˆ’ 0.0012407775638๏†1. Therefore ๐’œ(๐’ฎโ„’(2, 57)) = 476837158125000๏ฃ1. 4. Conclusion From our work we deduced that ๐’œ(๐’ฎโ„’(2, 5๐‘›)) where ๐‘› is an odd number and ๐‘› โ‰ฅ 3 equal to the order of the group and the number of the induced characters ฮฆ๐“ˆ equal to the number of the cyclic subgroups generated by the conjugacy classes of the group. Acknowledgment Our researcher extends his Sincere thanks to the editor and members of the preparatory committee of the Ibn AL-Haitham Journal of Pure and Applied Sciences. Conflict of Interest There are no conflicts of interest. Funding There is no funding for the article. References 1. Mohamed, S K. On Rational -Valued Characters of Certain Types of Permutation Group. Ibn Al-Haitham journal for pure and applied sciences 2006, 19(4), 99-108. 2. Saad, O. B. Investigating Particular Representations for Matrix Lie GroupsSO(3) andSL(2,โ‚ต). Iraqi journal of Science 2019, 60(4), 856-858, DOI: 10.24996/ijs.2019.60.4.19. 3. Sigler, L.E. Algebra, 1976, Springer-Verlage, Berlin. 4. Niran, S. J.; Hadeel H. L; Rana N. M. Computations for the special linear group (2,49). Journal of Interdisciplinary Mathematics 2021, 24(6), 1677โ€“1683, DOI: 10.1080/09720502.2021.1892273. 5. Mohammed I. L., Niran S. J., Issa A. Score for the group SL (2,38). Ibn Al-Haitham Journal for Pure and Applied Sciences 2023, 36(3), 408โ€“415, DOI:10.30526/36.3.3017. IHJPAS. 37 (1) 2024 411 6. Noor Alhuda, S. S.; Niran S. J. Periodical split for the groups PSL(2,31) and PSL(2,37). Journal of Discrete Mathematical Sciences and Cryptography 2022, 25(2), 605โ€“608, DOI : 10.1080/09720529.2021.1982490. 7. Sherouk, A. K.; Niran S. J. Calculation for the groups SL(2,U), U = 31 and 37. Journal of Discrete Mathematical Sciences and Cryptography 2022, 25(2), 609-613, DOI: 10.1080/09720529.2021.1972614. 8. Salih,O. M.; Mohammed,N. J.; Alwan,B. M. Result for the Groups ๐’ฎ๐’ฐ๐’ฏ(2, ๐“…), where ๐“… = 3, 5, 7. Technology reports of Kansai university 2020, 62 (3), 2029-2034. 9. Dunya, M. H.; Ahmed, K. M.; Intidhar, Z. M. Score for some Groups SUT(2,p). Int. J. Nonlinear Anal. Appl. 2021, 12(2), 1-15. 10. Maha, A. M.; Lemya A. Al. H.; Asma A. A. New algorithm based on deep learning for number recognition. International Journal of Mathematics and Computer Science 2023, 18( 3), 429โ€“438. 11. Anwar, K. F.; Lemya, A. Al. H.; Areej, M. A.; Shatha A. S. Symmetric generalized bi- derivations with prime ideals. International Journal of Mathematics and Computer Science 2023, 18(4), 675โ€“684,. 12. Bill, C. Representations of SL2 (R). Pacific Journal of Mathematics 2020, 3(1), 231-250. 13. Yurii, I. L. Introduction to the Theory of Banach Representations of Groups 2021, Birkhรคuser Verlag. 14. Chemistry, L. Representations of Groups 2019, Kharkov, Ukraine. 15. Zagier, D. Applications of the representation theory of finite groups 2019, New York. 16. Alexander, K. Jr. An introduction to Lie groups and Lie algebras 2022, Birkhรคuser Verlag. 17. Reiner, I. Representation theory 2021, John Wiley & Sons, NewYork โ€“London. 18. Kevin, H. on representation theory in groups, 2020, Springer-Verlage. 19. Mohammed, S. I.K.; Lemia, A. Al. H. The Artin' s Exponent of A Special Linear Group SL(2,2k ). Eng. & Tech. Journal 2010, 28(10), 1924-1934. 20. Yuxin, Z. T.; Orkesh, N.; Zhang, Z. Homology Algebra and Applications. J.Sci.I.R.Iran 2019, 3(5) 11-13. 21. Curtis, C.W. ; Reiner, I. The Representation Theory Of Finite Groups And Associative Algebras 1962, John Wiley & Sons, NewYork โ€“London. 22. Behravesh, H. The Rational Character Table Of Special Linear Groups. J.Sci.I.R.Iran 1998, 9 (2), 173-180. 23. Behravesh, H. Quasi โ€“ Permutation Representations Of SL(n,q) And PSL(n,q). Glasgow.Math.J. 1999, 41, 393- 408. 24. Isaacs, I.M. Character Theory Of Finite Groups 1976, Academic Press, NewYork. 25. Serre, J.P. Linear Representation Of Finite Groups 1977, Springer-Verlage. 26. Gehles, K.E. Ordinary Characters Of Finite Special Linear Groups 2002, M.Sc. Thesis, Dissertation, University of St Andrews. 27. Kirdar, M.S. The Factor Group Of The Z-Valued Class Function Modulo The Group Of The Generalized Characters 1982, Ph.D.Thesis, University of Birmingham. 28. Kirdar, M.S. On Brauerโ€™s Proof Of The Artin Induction Theorem. Abhath ALโ€“Yarmouk (Basic Sciences and Engineering) journal 2002, 11(1A), 51โ€“54. 29. Apostol, T.M. Introduction To Analytic Number Theory 1976, Springer-Verlage, NewYork. 30. Hall, B. C. Lie Groups, Lie Algebras, and Representations 2015, Springer. https://books.google.com/books?id=gCr3BwAAQBAJ&q=%22Introduction+to+the+Theory+of+Banach+Representations+of+Groups%22 https://en.wikipedia.org/wiki/Alexander_Kirillov_Jr. http://www.math.sunysb.edu/~kirillov/liegroups/ https://www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609/