Aliases: C8.5S4, U2(F3):5C2, GL2(F3):2C4, CSU2(F3):2C4, C8.A4:6C2, C8oD4:4S3, C2.9(C4xS4), C4.29(C2xS4), C4.6S4.C2, Q8.5(C4xS3), C4oD4.10D6, C4.A4.12C22, SL2(F3).4(C2xC4), SmallGroup(192,964)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(F3) — C8.5S4 |
Generators and relations for C8.5S4
G = < a,b,c,d,e | a8=d3=e2=1, b2=c2=a4, ab=ba, ac=ca, ad=da, eae=a5, cbc-1=a4b, dbd-1=a4bc, ebe=bc, dcd-1=b, ece=a4c, ede=d-1 >
Subgroups: 215 in 62 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2xC4, D4, Q8, Q8, Dic3, C12, D6, C42, C2xC8, M4(2), D8, SD16, Q16, C4oD4, C4oD4, C3:C8, C24, SL2(F3), C4xS3, C8:C4, C4wrC2, C8.C4, C8oD4, C8oD4, C4oD8, C8:S3, CSU2(F3), GL2(F3), C4.A4, C8.26D4, U2(F3), C8.A4, C4.6S4, C8.5S4
Quotients: C1, C2, C4, C22, S3, C2xC4, D6, C4xS3, S4, C2xS4, C4xS4, C8.5S4
Character table of C8.5S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 6 | 12 | 8 | 1 | 1 | 6 | 12 | 12 | 12 | 8 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | i | -i | -1 | 1 | -i | i | -i | i | i | -1 | -i | 1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | i | -1 | 1 | i | -i | i | -i | -i | -1 | i | 1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | 1 | 1 | -i | i | -i | i | -i | 1 | i | -1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | 1 | 1 | i | -i | i | -i | i | 1 | -i | -1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 1 | 1 | -i | i | -i | i | complex lifted from C4xS3 |
| ρ12 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 1 | 1 | i | -i | i | -i | complex lifted from C4xS3 |
| ρ13 | 3 | 3 | -1 | 1 | 0 | 3 | 3 | -1 | 1 | 1 | 1 | 0 | -3 | -3 | 1 | 1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ14 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ15 | 3 | 3 | -1 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | 0 | 3 | 3 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | 1 | 1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 1 | -1 | 0 | -3 | -3 | -1 | -i | i | 1 | 0 | 3i | -3i | -i | i | i | -1 | -i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4xS4 |
| ρ18 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | i | -i | -1 | 0 | 3i | -3i | -i | i | -i | 1 | i | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4xS4 |
| ρ19 | 3 | 3 | 1 | -1 | 0 | -3 | -3 | -1 | i | -i | 1 | 0 | -3i | 3i | i | -i | -i | -1 | i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4xS4 |
| ρ20 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | -i | i | -1 | 0 | -3i | 3i | i | -i | i | 1 | -i | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4xS4 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | 2ζ8ζ3+ζ8 | 2ζ87ζ3+ζ87 | 2ζ85ζ3+ζ85 | 2ζ83ζ3+ζ83 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | 2ζ87ζ3+ζ87 | 2ζ8ζ3+ζ8 | 2ζ83ζ3+ζ83 | 2ζ85ζ3+ζ85 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | 2ζ83ζ3+ζ83 | 2ζ85ζ3+ζ85 | 2ζ87ζ3+ζ87 | 2ζ8ζ3+ζ8 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | 2ζ85ζ3+ζ85 | 2ζ83ζ3+ζ83 | 2ζ8ζ3+ζ8 | 2ζ87ζ3+ζ87 | complex faithful |
►On 32 points
Generators in S32
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 14 5 10)(2 15 6 11)(3 16 7 12)(4 9 8 13)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 18 13 22)(10 19 14 23)(11 20 15 24)(12 21 16 17)
(9 28 22)(10 29 23)(11 30 24)(12 31 17)(13 32 18)(14 25 19)(15 26 20)(16 27 21)
(2 6)(4 8)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)(25 29)(27 31)
G:=sub<Sym(32)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17), (9,28,22)(10,29,23)(11,30,24)(12,31,17)(13,32,18)(14,25,19)(15,26,20)(16,27,21), (2,6)(4,8)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)(25,29)(27,31)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17), (9,28,22)(10,29,23)(11,30,24)(12,31,17)(13,32,18)(14,25,19)(15,26,20)(16,27,21), (2,6)(4,8)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)(25,29)(27,31) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,14,5,10),(2,15,6,11),(3,16,7,12),(4,9,8,13),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,18,13,22),(10,19,14,23),(11,20,15,24),(12,21,16,17)], [(9,28,22),(10,29,23),(11,30,24),(12,31,17),(13,32,18),(14,25,19),(15,26,20),(16,27,21)], [(2,6),(4,8),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17),(25,29),(27,31)]])
Matrix representation of C8.5S4 ►in GL4(F5) generated by
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 3 |
| 4 | 3 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 |
| 1 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 3 | 2 |
| 2 | 1 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 1 | 4 |
| 2 | 3 | 0 | 0 |
| 1 | 2 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 2 |
| 1 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,0,4,3,0,0,3,0,0,4,0,0,4,3,0,0],[1,1,0,0,3,4,0,0,0,0,3,3,0,0,0,2],[2,0,0,0,1,3,0,0,0,0,1,1,0,0,3,4],[2,1,0,0,3,2,0,0,0,0,1,3,0,0,4,3],[0,0,1,4,0,0,0,3,1,2,0,0,0,2,0,0] >;
C8.5S4 in GAP, Magma, Sage, TeX
C_8._5S_4
% in TeX
G:=Group("C8.5S4"); // GroupNames label
G:=SmallGroup(192,964);
// by ID
G=gap.SmallGroup(192,964);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,1373,36,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=d^3=e^2=1,b^2=c^2=a^4,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^5,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^4*c,e*d*e=d^-1>;
// generators/relations
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