Aliases: C8.3S4, Q8.5D12, SL2(𝔽3).5D4, C8.A4⋊1C2, C8○D4⋊2S3, C4.21(C2×S4), C2.11(C4⋊S4), C4.3S4⋊1C2, C4○D4.12D6, C4.A4.9C22, SmallGroup(192,966)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.3S4
G = < a,b,c,d,e | a8=d3=e2=1, b2=c2=a4, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a4b, dbd-1=a4bc, ebe=bc, dcd-1=b, ece=a4c, ede=d-1 >
Subgroups: 357 in 65 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, C23, C12, D6, C2×C8, M4(2), D8, SD16, C2×D4, C4○D4, C24, SL2(𝔽3), D12, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, D24, GL2(𝔽3), C4.A4, D4.4D4, C8.A4, C4.3S4, C8.3S4
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2×S4, C4⋊S4, C8.3S4
Character table of C8.3S4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 6 | 24 | 24 | 8 | 2 | 6 | 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 | 0 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ10 | 3 | 3 | -1 | -1 | -1 | 0 | 3 | -1 | 0 | 3 | 3 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ11 | 3 | 3 | -1 | 1 | -1 | 0 | 3 | -1 | 0 | -3 | -3 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ12 | 3 | 3 | -1 | -1 | 1 | 0 | 3 | -1 | 0 | -3 | -3 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ13 | 3 | 3 | -1 | 1 | 1 | 0 | 3 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal faithful |
| ρ15 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 0 | -√3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | orthogonal faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 0 | √3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | orthogonal faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | -2√2 | 2√2 | 0 | 0 | 0 | √3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | orthogonal faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | 2√2 | -2√2 | 0 | 0 | 0 | -√3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | orthogonal faithful |
| ρ20 | 6 | 6 | 2 | 0 | 0 | 0 | -6 | -2 | 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 11 5 15)(2 12 6 16)(3 13 7 9)(4 14 8 10)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(9 26 21)(10 27 22)(11 28 23)(12 29 24)(13 30 17)(14 31 18)(15 32 19)(16 25 20)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 26)(27 32)(28 31)(29 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), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (9,26,21)(10,27,22)(11,28,23)(12,29,24)(13,30,17)(14,31,18)(15,32,19)(16,25,20), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,26)(27,32)(28,31)(29,30)>;
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), (1,11,5,15)(2,12,6,16)(3,13,7,9)(4,14,8,10)(17,26,21,30)(18,27,22,31)(19,28,23,32)(20,29,24,25), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (9,26,21)(10,27,22)(11,28,23)(12,29,24)(13,30,17)(14,31,18)(15,32,19)(16,25,20), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)(25,26)(27,32)(28,31)(29,30) );
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)], [(1,11,5,15),(2,12,6,16),(3,13,7,9),(4,14,8,10),(17,26,21,30),(18,27,22,31),(19,28,23,32),(20,29,24,25)], [(1,28,5,32),(2,29,6,25),(3,30,7,26),(4,31,8,27),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24)], [(9,26,21),(10,27,22),(11,28,23),(12,29,24),(13,30,17),(14,31,18),(15,32,19),(16,25,20)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,24),(14,23),(15,22),(16,21),(25,26),(27,32),(28,31),(29,30)]])
Matrix representation of C8.3S4 ►in GL4(𝔽7) generated by
| 1 | 4 | 0 | 5 |
| 6 | 1 | 6 | 6 |
| 5 | 5 | 5 | 4 |
| 4 | 3 | 5 | 1 |
| 0 | 1 | 2 | 3 |
| 5 | 3 | 3 | 6 |
| 3 | 2 | 6 | 0 |
| 3 | 0 | 2 | 5 |
| 3 | 2 | 0 | 6 |
| 3 | 2 | 4 | 5 |
| 6 | 4 | 2 | 0 |
| 2 | 3 | 1 | 0 |
| 5 | 3 | 4 | 0 |
| 4 | 2 | 2 | 0 |
| 5 | 1 | 3 | 0 |
| 3 | 5 | 1 | 2 |
| 4 | 5 | 0 | 1 |
| 6 | 1 | 6 | 6 |
| 5 | 6 | 1 | 4 |
| 4 | 3 | 5 | 1 |
G:=sub<GL(4,GF(7))| [1,6,5,4,4,1,5,3,0,6,5,5,5,6,4,1],[0,5,3,3,1,3,2,0,2,3,6,2,3,6,0,5],[3,3,6,2,2,2,4,3,0,4,2,1,6,5,0,0],[5,4,5,3,3,2,1,5,4,2,3,1,0,0,0,2],[4,6,5,4,5,1,6,3,0,6,1,5,1,6,4,1] >;
C8.3S4 in GAP, Magma, Sage, TeX
C_8._3S_4
% in TeX
G:=Group("C8.3S4"); // GroupNames label
G:=SmallGroup(192,966);
// by ID
G=gap.SmallGroup(192,966);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,708,2102,520,451,1684,655,172,1013,404,285,124]);
// 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^-1,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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