Aliases: C8.4S4, Q8.4D12, SL2(𝔽3).4D4, C8○D4⋊1S3, C8.A4⋊2C2, C4.20(C2×S4), C4.3S4.C2, C2.10(C4⋊S4), C4.S4⋊1C2, C4○D4.11D6, 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, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C24, SL2(𝔽3), Dic6, D12, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, 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, C2×S4, 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 C2×S4 |
| ρ13 | 3 | 3 | -1 | -1 | 0 | 3 | -1 | 1 | 0 | -3 | -3 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| 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 |
►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 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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