p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C4⋊C8, C4.4Q8, C4.18D4, C42.2C2, C2.3M4(2), (C2×C4).4C4, C2.2(C2×C8), (C2×C8).2C2, C2.2(C4⋊C4), C22.9(C2×C4), (C2×C4).33C22, SmallGroup(32,12)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊C8
G = < a,b | a4=b8=1, bab-1=a-1 >
Character table of C4⋊C8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -i | i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | i | -i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | -1 | i | 1 | ζ85 | ζ87 | ζ8 | ζ83 | ζ87 | ζ85 | ζ8 | ζ83 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | -i | -i | i | i | i | 1 | -i | -1 | ζ8 | ζ83 | ζ85 | ζ87 | ζ87 | ζ85 | ζ8 | ζ83 | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | -1 | i | 1 | ζ8 | ζ83 | ζ85 | ζ87 | ζ83 | ζ8 | ζ85 | ζ87 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | -i | -i | i | i | i | 1 | -i | -1 | ζ85 | ζ87 | ζ8 | ζ83 | ζ83 | ζ8 | ζ85 | ζ87 | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | i | i | -i | -i | -i | 1 | i | -1 | ζ83 | ζ8 | ζ87 | ζ85 | ζ85 | ζ87 | ζ83 | ζ8 | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | i | i | -i | -i | -i | 1 | i | -1 | ζ87 | ζ85 | ζ83 | ζ8 | ζ8 | ζ83 | ζ87 | ζ85 | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | -1 | -i | 1 | ζ87 | ζ85 | ζ83 | ζ8 | ζ85 | ζ87 | ζ83 | ζ8 | linear of order 8 |
| ρ16 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | -1 | -i | 1 | ζ83 | ζ8 | ζ87 | ζ85 | ζ8 | ζ83 | ζ87 | ζ85 | linear of order 8 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | -2 | 2 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
►Regular action on 32 points
(1 23 31 12)(2 13 32 24)(3 17 25 14)(4 15 26 18)(5 19 27 16)(6 9 28 20)(7 21 29 10)(8 11 30 22)
(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)
G:=sub<Sym(32)| (1,23,31,12)(2,13,32,24)(3,17,25,14)(4,15,26,18)(5,19,27,16)(6,9,28,20)(7,21,29,10)(8,11,30,22), (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)>;
G:=Group( (1,23,31,12)(2,13,32,24)(3,17,25,14)(4,15,26,18)(5,19,27,16)(6,9,28,20)(7,21,29,10)(8,11,30,22), (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) );
G=PermutationGroup([[(1,23,31,12),(2,13,32,24),(3,17,25,14),(4,15,26,18),(5,19,27,16),(6,9,28,20),(7,21,29,10),(8,11,30,22)], [(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)]])
C4⋊C8 is a maximal subgroup of
C4.D8 C4.10D8 C4.6Q16 C4⋊M4(2) C42.6C22 C42.7C22 C8×D4 C8⋊9D4 C8⋊6D4 C8×Q8 C4⋊D8 C4⋊SD16 D4.D4 C4⋊2Q16 D4.2D4 Q8.D4 D4⋊Q8 Q8⋊Q8 D4⋊2Q8 C4.Q16 D4.Q8 Q8.Q8 C32⋊C4⋊C8 C4.4PSU3(𝔽2) (C3×C12)⋊4C8 C32⋊5(C4⋊C8) C4⋊F9
C4p⋊C8: C8⋊2C8 C8⋊1C8 C12⋊C8 C20⋊3C8 C20⋊C8 C28⋊C8 C44⋊C8 C52⋊3C8 ...
C2p.M4(2): D4⋊C8 Q8⋊C8 C42.12C4 C42.6C4 C8⋊4Q8 Dic3⋊C8 C20.8Q8 Dic5⋊C8 ...
C4⋊C8 is a maximal quotient of
C22.7C42 Dic5⋊C8 C32⋊C4⋊C8 C4.4PSU3(𝔽2) (C3×C12)⋊4C8 C32⋊5(C4⋊C8) C4⋊F9 Dic13⋊C8
C4p⋊C8: C8⋊2C8 C8⋊1C8 C12⋊C8 C20⋊3C8 C20⋊C8 C28⋊C8 C44⋊C8 C52⋊3C8 ...
C4p.Q8: C4⋊C16 C8.C8 Dic3⋊C8 C20.8Q8 Dic7⋊C8 Dic11⋊C8 C52.8Q8 ...
Matrix representation of C4⋊C8 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 16 | 15 |
| 0 | 1 | 1 |
| 2 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 1 | 1 |
G:=sub<GL(3,GF(17))| [1,0,0,0,16,1,0,15,1],[2,0,0,0,16,1,0,0,1] >;
C4⋊C8 in GAP, Magma, Sage, TeX
C_4\rtimes C_8
% in TeX
G:=Group("C4:C8"); // GroupNames label
G:=SmallGroup(32,12);
// by ID
G=gap.SmallGroup(32,12);
# by ID
G:=PCGroup([5,-2,2,-2,2,-2,40,61,26,58]);
// Polycyclic
G:=Group<a,b|a^4=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C4⋊C8 in TeX
Character table of C4⋊C8 in TeX