Aliases: C8.2S4, Q8.3D12, SL2(F3).3D4, C2.9(C4:S4), C4.19(C2xS4), C4.S4.C2, C8oD4.1S3, C8.A4.1C2, C4oD4.8D6, C4.A4.7C22, SmallGroup(192,962)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.S4
G = < a,b,c,d,e | a8=d3=1, b2=c2=e2=a4, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a4b, dbd-1=a4bc, ebe-1=bc, dcd-1=b, ece-1=a4c, ede-1=d-1 >
Subgroups: 229 in 59 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C8, C2xC4, D4, Q8, Q8, Dic3, C12, C2xC8, M4(2), SD16, Q16, C2xQ8, C4oD4, C24, SL2(F3), Dic6, C4.10D4, C8.C4, C8oD4, C2xQ16, C8.C22, Dic12, CSU2(F3), C4.A4, D4.5D4, C8.A4, C4.S4, C8.S4
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2xS4, C4:S4, C8.S4
Character table of C8.S4
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 6 | 8 | 2 | 6 | 24 | 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 | 2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ9 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ10 | 3 | 3 | -1 | 0 | 3 | -1 | -1 | 1 | 0 | -3 | -3 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ11 | 3 | 3 | -1 | 0 | 3 | -1 | 1 | -1 | 0 | -3 | -3 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ12 | 3 | 3 | -1 | 0 | 3 | -1 | 1 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ13 | 3 | 3 | -1 | 0 | 3 | -1 | -1 | -1 | 0 | 3 | 3 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic faithful, Schur index 2 |
| ρ15 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic faithful, Schur index 2 |
| ρ16 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 0 | -√3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | symplectic faithful, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 0 | √3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | symplectic faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 0 | -√3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | symplectic faithful, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 0 | √3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | symplectic faithful, Schur index 2 |
| ρ20 | 6 | 6 | 2 | 0 | -6 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4:S4 |
►On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(9 61 13 57)(10 62 14 58)(11 63 15 59)(12 64 16 60)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 32 13 28)(10 25 14 29)(11 26 15 30)(12 27 16 31)(17 64 21 60)(18 57 22 61)(19 58 23 62)(20 59 24 63)(41 49 45 53)(42 50 46 54)(43 51 47 55)(44 52 48 56)
(17 64 27)(18 57 28)(19 58 29)(20 59 30)(21 60 31)(22 61 32)(23 62 25)(24 63 26)(33 43 51)(34 44 52)(35 45 53)(36 46 54)(37 47 55)(38 48 56)(39 41 49)(40 42 50)
(1 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 56 21 52)(18 55 22 51)(19 54 23 50)(20 53 24 49)(25 40 29 36)(26 39 30 35)(27 38 31 34)(28 37 32 33)(41 59 45 63)(42 58 46 62)(43 57 47 61)(44 64 48 60)
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,61,13,57)(10,62,14,58)(11,63,15,59)(12,64,16,60)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (17,64,27)(18,57,28)(19,58,29)(20,59,30)(21,60,31)(22,61,32)(23,62,25)(24,63,26)(33,43,51)(34,44,52)(35,45,53)(36,46,54)(37,47,55)(38,48,56)(39,41,49)(40,42,50), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,59,45,63)(42,58,46,62)(43,57,47,61)(44,64,48,60)>;
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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(9,61,13,57)(10,62,14,58)(11,63,15,59)(12,64,16,60)(17,31,21,27)(18,32,22,28)(19,25,23,29)(20,26,24,30)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,32,13,28)(10,25,14,29)(11,26,15,30)(12,27,16,31)(17,64,21,60)(18,57,22,61)(19,58,23,62)(20,59,24,63)(41,49,45,53)(42,50,46,54)(43,51,47,55)(44,52,48,56), (17,64,27)(18,57,28)(19,58,29)(20,59,30)(21,60,31)(22,61,32)(23,62,25)(24,63,26)(33,43,51)(34,44,52)(35,45,53)(36,46,54)(37,47,55)(38,48,56)(39,41,49)(40,42,50), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,56,21,52)(18,55,22,51)(19,54,23,50)(20,53,24,49)(25,40,29,36)(26,39,30,35)(27,38,31,34)(28,37,32,33)(41,59,45,63)(42,58,46,62)(43,57,47,61)(44,64,48,60) );
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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,53,5,49),(2,54,6,50),(3,55,7,51),(4,56,8,52),(9,61,13,57),(10,62,14,58),(11,63,15,59),(12,64,16,60),(17,31,21,27),(18,32,22,28),(19,25,23,29),(20,26,24,30),(33,43,37,47),(34,44,38,48),(35,45,39,41),(36,46,40,42)], [(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,32,13,28),(10,25,14,29),(11,26,15,30),(12,27,16,31),(17,64,21,60),(18,57,22,61),(19,58,23,62),(20,59,24,63),(41,49,45,53),(42,50,46,54),(43,51,47,55),(44,52,48,56)], [(17,64,27),(18,57,28),(19,58,29),(20,59,30),(21,60,31),(22,61,32),(23,62,25),(24,63,26),(33,43,51),(34,44,52),(35,45,53),(36,46,54),(37,47,55),(38,48,56),(39,41,49),(40,42,50)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,56,21,52),(18,55,22,51),(19,54,23,50),(20,53,24,49),(25,40,29,36),(26,39,30,35),(27,38,31,34),(28,37,32,33),(41,59,45,63),(42,58,46,62),(43,57,47,61),(44,64,48,60)]])
Matrix representation of C8.S4 ►in GL4(F7) generated by
| 1 | 4 | 0 | 5 |
| 6 | 1 | 6 | 6 |
| 5 | 5 | 5 | 4 |
| 4 | 3 | 5 | 1 |
| 3 | 2 | 0 | 6 |
| 3 | 2 | 4 | 5 |
| 6 | 4 | 2 | 0 |
| 2 | 3 | 1 | 0 |
| 0 | 2 | 3 | 2 |
| 2 | 3 | 4 | 4 |
| 5 | 1 | 1 | 0 |
| 4 | 6 | 5 | 3 |
| 5 | 2 | 3 | 1 |
| 0 | 2 | 1 | 2 |
| 3 | 2 | 1 | 0 |
| 2 | 6 | 5 | 4 |
| 6 | 6 | 2 | 0 |
| 6 | 0 | 1 | 4 |
| 2 | 3 | 5 | 2 |
| 1 | 4 | 1 | 3 |
G:=sub<GL(4,GF(7))| [1,6,5,4,4,1,5,3,0,6,5,5,5,6,4,1],[3,3,6,2,2,2,4,3,0,4,2,1,6,5,0,0],[0,2,5,4,2,3,1,6,3,4,1,5,2,4,0,3],[5,0,3,2,2,2,2,6,3,1,1,5,1,2,0,4],[6,6,2,1,6,0,3,4,2,1,5,1,0,4,2,3] >;
C8.S4 in GAP, Magma, Sage, TeX
C_8.S_4
% in TeX
G:=Group("C8.S4"); // GroupNames label
G:=SmallGroup(192,962);
// by ID
G=gap.SmallGroup(192,962);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,708,2102,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=d^3=1,b^2=c^2=e^2=a^4,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^4*c,e*d*e^-1=d^-1>;
// generators/relations
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