Aliases: C8.4S4, Q8.4D12, SL2(F3).4D4, C8oD4:1S3, C8.A4:2C2, C4.20(C2xS4), C4.3S4.C2, C2.10(C4:S4), C4.S4:1C2, C4oD4.11D6, C4.A4.8C22, SmallGroup(192,965)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.4S4
G = < a,b,c,d,e | a8=d3=e2=1, b2=c2=a4, ab=ba, ac=ca, ad=da, eae=a3, cbc-1=a4b, dbd-1=a4bc, ebe=bc, dcd-1=b, ece=a4c, ede=d-1 >
Subgroups: 293 in 62 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2xC4, D4, Q8, Q8, C23, Dic3, C12, D6, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, C24, SL2(F3), Dic6, D12, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, C24:C2, CSU2(F3), GL2(F3), C4.A4, D4.3D4, C8.A4, C4.S4, C4.3S4, C8.4S4
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2xS4, C4:S4, C8.4S4
Character table of C8.4S4
class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 24A | 24B | 24C | 24D | |
size | 1 | 1 | 6 | 24 | 8 | 2 | 6 | 24 | 8 | 2 | 2 | 12 | 24 | 24 | 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 | trivial |
ρ2 | 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 | 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 | linear of order 2 |
ρ5 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ6 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ7 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ8 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
ρ9 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
ρ10 | 3 | 3 | -1 | 1 | 0 | 3 | -1 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
ρ11 | 3 | 3 | -1 | -1 | 0 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
ρ12 | 3 | 3 | -1 | 1 | 0 | 3 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
ρ13 | 3 | 3 | -1 | -1 | 0 | 3 | -1 | 1 | 0 | -3 | -3 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
ρ14 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | complex faithful |
ρ15 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | complex faithful |
ρ16 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | -2√-2 | 2√-2 | 0 | 0 | 0 | √3 | -√3 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | complex faithful |
ρ17 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | 2√-2 | -2√-2 | 0 | 0 | 0 | √3 | -√3 | ζ87ζ3+ζ87-ζ85ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | complex faithful |
ρ18 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | -2√-2 | 2√-2 | 0 | 0 | 0 | -√3 | √3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | complex faithful |
ρ19 | 4 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | 2√-2 | -2√-2 | 0 | 0 | 0 | -√3 | √3 | ζ87ζ32+ζ87-ζ85ζ32 | ζ87ζ3+ζ87-ζ85ζ3 | ζ83ζ32+ζ83-ζ8ζ32 | ζ83ζ3+ζ83-ζ8ζ3 | complex faithful |
ρ20 | 6 | 6 | 2 | 0 | 0 | -6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4:S4 |
(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 4)(3 7)(6 8)(9 24)(10 19)(11 22)(12 17)(13 20)(14 23)(15 18)(16 21)(25 29)(26 32)(28 30)
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), (2,4)(3,7)(6,8)(9,24)(10,19)(11,22)(12,17)(13,20)(14,23)(15,18)(16,21)(25,29)(26,32)(28,30)>;
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), (2,4)(3,7)(6,8)(9,24)(10,19)(11,22)(12,17)(13,20)(14,23)(15,18)(16,21)(25,29)(26,32)(28,30) );
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)], [(2,4),(3,7),(6,8),(9,24),(10,19),(11,22),(12,17),(13,20),(14,23),(15,18),(16,21),(25,29),(26,32),(28,30)]])
Matrix representation of C8.4S4 ►in GL4(F3) generated by
0 | 1 | 2 | 2 |
0 | 0 | 0 | 1 |
2 | 0 | 2 | 1 |
0 | 1 | 0 | 2 |
0 | 1 | 1 | 2 |
0 | 1 | 2 | 0 |
0 | 2 | 2 | 0 |
1 | 0 | 1 | 0 |
2 | 2 | 2 | 1 |
0 | 1 | 0 | 2 |
2 | 2 | 1 | 2 |
0 | 2 | 0 | 2 |
2 | 1 | 1 | 2 |
1 | 2 | 1 | 2 |
0 | 1 | 0 | 1 |
2 | 0 | 1 | 0 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
1 | 0 | 2 | 0 |
0 | 2 | 0 | 2 |
G:=sub<GL(4,GF(3))| [0,0,2,0,1,0,0,1,2,0,2,0,2,1,1,2],[0,0,0,1,1,1,2,0,1,2,2,1,2,0,0,0],[2,0,2,0,2,1,2,2,2,0,1,0,1,2,2,2],[2,1,0,2,1,2,1,0,1,1,0,1,2,2,1,0],[1,0,1,0,0,1,0,2,0,0,2,0,0,0,0,2] >;
C8.4S4 in GAP, Magma, Sage, TeX
C_8._4S_4
% in TeX
G:=Group("C8.4S4");
// GroupNames label
G:=SmallGroup(192,965);
// by ID
G=gap.SmallGroup(192,965);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,36,2102,520,451,1684,655,172,1013,404,285,124]);
// 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^3,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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