metabelian, soluble, monomial, A-group
Aliases: C32⋊2C8, (C3×C6).C4, C2.(C32⋊C4), C3⋊Dic3.1C2, SmallGroup(72,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32⋊2C8 |
Generators and relations for C32⋊2C8
G = < a,b,c | a3=b3=c8=1, cbc-1=ab=ba, cac-1=a-1b >
Character table of C32⋊2C8
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 4 | 4 | 9 | 9 | 4 | 4 | 9 | 9 | 9 | 9 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ5 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | ζ87 | ζ85 | ζ83 | ζ8 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | ζ8 | ζ83 | ζ85 | ζ87 | linear of order 8 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | ζ83 | ζ8 | ζ87 | ζ85 | linear of order 8 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | ζ85 | ζ87 | ζ8 | ζ83 | linear of order 8 |
| ρ9 | 4 | 4 | 1 | -2 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ10 | 4 | 4 | -2 | 1 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ11 | 4 | -4 | -2 | 1 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ12 | 4 | -4 | 1 | -2 | 0 | 0 | 2 | -1 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 24 points - transitive group 24T63
Generators in S24
(1 21 9)(2 10 22)(3 11 23)(4 24 12)(5 17 13)(6 14 18)(7 15 19)(8 20 16)
(2 22 10)(4 12 24)(6 18 14)(8 16 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,21,9)(2,10,22)(3,11,23)(4,24,12)(5,17,13)(6,14,18)(7,15,19)(8,20,16), (2,22,10)(4,12,24)(6,18,14)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (1,21,9)(2,10,22)(3,11,23)(4,24,12)(5,17,13)(6,14,18)(7,15,19)(8,20,16), (2,22,10)(4,12,24)(6,18,14)(8,16,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,21,9),(2,10,22),(3,11,23),(4,24,12),(5,17,13),(6,14,18),(7,15,19),(8,20,16)], [(2,22,10),(4,12,24),(6,18,14),(8,16,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,63);
C32⋊2C8 is a maximal subgroup of
C2.F9 C32⋊D8 C32⋊2SD16 C32⋊Q16 C3⋊S3⋊3C8 C32⋊M4(2) C62.C4 C33⋊4C8 (C3×C15)⋊9C8 (C3×C6).F5
C32⋊2C8 is a maximal quotient of
C32⋊2C16 He3⋊2C8 C33⋊4C8 (C3×C15)⋊9C8 (C3×C6).F5
Matrix representation of C32⋊2C8 ►in GL4(𝔽5) generated by
| 4 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 2 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [4,0,2,0,0,0,0,3,2,0,0,0,0,3,0,4],[1,0,0,0,0,4,0,2,0,0,1,0,0,2,0,0],[0,1,0,0,2,0,0,0,0,0,0,1,1,0,3,0] >;
C32⋊2C8 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2C_8
% in TeX
G:=Group("C3^2:2C8"); // GroupNames label
G:=SmallGroup(72,19);
// by ID
G=gap.SmallGroup(72,19);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,3,10,26,1123,168,1604,609]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^8=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b>;
// generators/relations
Export
Subgroup lattice of C32⋊2C8 in TeX
Character table of C32⋊2C8 in TeX