p-group, metabelian, nilpotent (class 4), monomial
Aliases: C42.1C8, C23.16M4(2), (C2×C8).17D4, (C2×C42).10C4, C23.C8.2C2, C4.41(C23⋊C4), (C2×M4(2)).8C4, C4.13(C4.10D4), C4⋊M4(2).11C2, C22.20(C22⋊C8), (C2×M4(2)).145C22, C2.8(C22.M4(2)), (C2×C4).36(C2×C8), (C22×C4).62(C2×C4), (C2×C4).344(C22⋊C4), SmallGroup(128,59)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.C8
G = < a,b,c | a4=b4=1, c8=b2, ab=ba, cac-1=a-1b-1, cbc-1=a2b >
Character table of C42.C8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | 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 | -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 | -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 | 1 | -1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -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 | -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 | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -i | i | ζ87 | ζ83 | ζ85 | ζ8 | ζ8 | ζ85 | ζ87 | ζ83 | linear of order 8 |
| ρ10 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | i | -i | -i | -i | i | ζ85 | ζ8 | ζ87 | ζ87 | ζ83 | ζ83 | ζ8 | ζ85 | linear of order 8 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -i | i | ζ83 | ζ87 | ζ8 | ζ85 | ζ85 | ζ8 | ζ83 | ζ87 | linear of order 8 |
| ρ12 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | i | -i | ζ8 | ζ85 | ζ83 | ζ87 | ζ87 | ζ83 | ζ8 | ζ85 | linear of order 8 |
| ρ13 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | i | -i | -i | -i | i | ζ8 | ζ85 | ζ83 | ζ83 | ζ87 | ζ87 | ζ85 | ζ8 | linear of order 8 |
| ρ14 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | -i | i | i | i | -i | ζ83 | ζ87 | ζ8 | ζ8 | ζ85 | ζ85 | ζ87 | ζ83 | linear of order 8 |
| ρ15 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | i | -i | ζ85 | ζ8 | ζ87 | ζ83 | ζ83 | ζ87 | ζ85 | ζ8 | linear of order 8 |
| ρ16 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | -i | i | i | i | -i | ζ87 | ζ83 | ζ85 | ζ85 | ζ8 | ζ8 | ζ83 | ζ87 | linear of order 8 |
| ρ17 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 4 | 4 | -4 | 0 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ22 | 4 | 4 | -4 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | -2 | 2i | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | -2 | -2i | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | -4i | 4i | 0 | 2 | 2i | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 4i | -4i | 0 | 2 | -2i | 2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T242
Generators in S16
(1 13 9 5)(2 10)(3 7 11 15)(4 12)(6 14)(8 16)
(1 13 9 5)(2 6 10 14)(3 7 11 15)(4 16 12 8)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,13,9,5)(2,10)(3,7,11,15)(4,12)(6,14)(8,16), (1,13,9,5)(2,6,10,14)(3,7,11,15)(4,16,12,8), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)>;
G:=Group( (1,13,9,5)(2,10)(3,7,11,15)(4,12)(6,14)(8,16), (1,13,9,5)(2,6,10,14)(3,7,11,15)(4,16,12,8), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,13,9,5),(2,10),(3,7,11,15),(4,12),(6,14),(8,16)], [(1,13,9,5),(2,6,10,14),(3,7,11,15),(4,16,12,8)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,242);
Matrix representation of C42.C8 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 2 |
| 0 | 0 | 4 | 0 |
| 0 | 4 | 0 | 1 |
| 4 | 0 | 2 | 0 |
| 0 | 4 | 0 | 2 |
| 4 | 0 | 1 | 0 |
| 0 | 4 | 0 | 1 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,4,0,0,4,0,0,2,0,1],[4,0,4,0,0,4,0,4,2,0,1,0,0,2,0,1],[0,1,0,0,0,0,1,0,0,0,0,1,2,0,0,0] >;
C42.C8 in GAP, Magma, Sage, TeX
C_4^2.C_8
% in TeX
G:=Group("C4^2.C8"); // GroupNames label
G:=SmallGroup(128,59);
// by ID
G=gap.SmallGroup(128,59);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,352,1242,521,136,2804,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^8=b^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^2*b>;
// generators/relations
Export
Subgroup lattice of C42.C8 in TeX
Character table of C42.C8 in TeX