p-group, metabelian, nilpotent (class 5), monomial
Aliases: (C4×C8)⋊3C4, (C2×D4).8D4, (C2×Q8).8D4, C42.C2⋊2C4, C42.19(C2×C4), C42⋊3C4.2C2, C2.8(C42⋊3C4), C42.C4.2C2, C4.4D4.5C22, C22.24(C23⋊C4), C42.78C22.1C2, (C2×C4).40(C22⋊C4), SmallGroup(128,146)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C4×C8)⋊C4
G = < a,b,c | a4=b8=c4=1, ab=ba, cac-1=a-1b2, cbc-1=a-1b-1 >
Character table of (C4×C8)⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 2 | 8 | 4 | 4 | 4 | 8 | 16 | 16 | 16 | 4 | 4 | 4 | 4 | 16 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -i | 1 | i | -1 | -1 | -1 | -1 | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | i | -1 | -i | 1 | 1 | 1 | 1 | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -i | -1 | i | 1 | 1 | 1 | 1 | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | i | 1 | -i | -1 | -1 | -1 | -1 | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ12 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | complex lifted from C42⋊3C4 |
| ρ13 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | complex lifted from C42⋊3C4 |
| ρ14 | 4 | -4 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 2ζ8 | 2ζ85 | 2ζ83 | 2ζ87 | 0 | 0 | complex faithful |
| ρ15 | 4 | -4 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 2ζ85 | 2ζ8 | 2ζ87 | 2ζ83 | 0 | 0 | complex faithful |
| ρ16 | 4 | -4 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 2ζ83 | 2ζ87 | 2ζ8 | 2ζ85 | 0 | 0 | complex faithful |
| ρ17 | 4 | -4 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 2ζ87 | 2ζ83 | 2ζ85 | 2ζ8 | 0 | 0 | complex faithful |
►On 32 points
Generators in S32
(1 12 27 20)(2 13 28 21)(3 14 29 22)(4 15 30 23)(5 16 31 24)(6 9 32 17)(7 10 25 18)(8 11 26 19)
(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)
(2 17 26 19)(3 29 7 25)(4 11 32 9)(6 21 30 23)(8 15 28 13)(10 20 18 16)(12 14 24 22)(27 31)
G:=sub<Sym(32)| (1,12,27,20)(2,13,28,21)(3,14,29,22)(4,15,30,23)(5,16,31,24)(6,9,32,17)(7,10,25,18)(8,11,26,19), (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), (2,17,26,19)(3,29,7,25)(4,11,32,9)(6,21,30,23)(8,15,28,13)(10,20,18,16)(12,14,24,22)(27,31)>;
G:=Group( (1,12,27,20)(2,13,28,21)(3,14,29,22)(4,15,30,23)(5,16,31,24)(6,9,32,17)(7,10,25,18)(8,11,26,19), (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), (2,17,26,19)(3,29,7,25)(4,11,32,9)(6,21,30,23)(8,15,28,13)(10,20,18,16)(12,14,24,22)(27,31) );
G=PermutationGroup([[(1,12,27,20),(2,13,28,21),(3,14,29,22),(4,15,30,23),(5,16,31,24),(6,9,32,17),(7,10,25,18),(8,11,26,19)], [(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)], [(2,17,26,19),(3,29,7,25),(4,11,32,9),(6,21,30,23),(8,15,28,13),(10,20,18,16),(12,14,24,22),(27,31)]])
Matrix representation of (C4×C8)⋊C4 ►in GL4(𝔽17) generated by
| 2 | 15 | 15 | 15 |
| 15 | 2 | 15 | 15 |
| 2 | 2 | 2 | 15 |
| 2 | 2 | 15 | 2 |
| 13 | 1 | 1 | 13 |
| 1 | 13 | 13 | 1 |
| 16 | 4 | 13 | 1 |
| 4 | 16 | 1 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [2,15,2,2,15,2,2,2,15,15,2,15,15,15,15,2],[13,1,16,4,1,13,4,16,1,13,13,1,13,1,1,13],[1,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0] >;
(C4×C8)⋊C4 in GAP, Magma, Sage, TeX
(C_4\times C_8)\rtimes C_4
% in TeX
G:=Group("(C4xC8):C4"); // GroupNames label
G:=SmallGroup(128,146);
// by ID
G=gap.SmallGroup(128,146);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,232,422,1059,184,1690,745,1684,1411,375,172,4037]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a^-1*b^-1>;
// generators/relations
Export
Subgroup lattice of (C4×C8)⋊C4 in TeX
Character table of (C4×C8)⋊C4 in TeX