direct product, p-group, abelian, monomial
Aliases: C4×C8, SmallGroup(32,3)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C4×C8 |
| C1 — C4×C8 |
| C1 — C4×C8 |
Generators and relations for C4×C8
G = < a,b | a4=b8=1, ab=ba >
Smallest permutation representation of C4×C8
►Regular action on 32 points
Generators in S32
(1 21 31 14)(2 22 32 15)(3 23 25 16)(4 24 26 9)(5 17 27 10)(6 18 28 11)(7 19 29 12)(8 20 30 13)
(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)| (1,21,31,14)(2,22,32,15)(3,23,25,16)(4,24,26,9)(5,17,27,10)(6,18,28,11)(7,19,29,12)(8,20,30,13), (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( (1,21,31,14)(2,22,32,15)(3,23,25,16)(4,24,26,9)(5,17,27,10)(6,18,28,11)(7,19,29,12)(8,20,30,13), (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([[(1,21,31,14),(2,22,32,15),(3,23,25,16),(4,24,26,9),(5,17,27,10),(6,18,28,11),(7,19,29,12),(8,20,30,13)], [(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
C8⋊C8 D4⋊C8 Q8⋊C8 C8⋊2C8 C8⋊1C8 C16⋊5C4 C4⋊C16 C8.C8 C8○2M4(2) C42.12C4 C42.7C22 C8⋊6D4 C8○D8 C8⋊4Q8 C4.4D8 C4.SD16 C42.78C22 C8⋊5D4 C8⋊4D4 C4⋊Q16 C8.12D4 C8⋊3Q8 C8.5Q8 C8⋊2Q8
C4×C8 is a maximal quotient of
C8⋊C8 C22.7C42 C16⋊5C4
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||
| image | C1 | C2 | C2 | C4 | C4 | C8 |
| kernel | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 4 | 16 |
Matrix representation of C4×C8 ►in GL2(𝔽17) generated by
| 4 | 0 |
| 0 | 4 |
| 2 | 0 |
| 0 | 4 |
G:=sub<GL(2,GF(17))| [4,0,0,4],[2,0,0,4] >;
C4×C8 in GAP, Magma, Sage, TeX
C_4\times C_8
% in TeX
G:=Group("C4xC8"); // GroupNames label
G:=SmallGroup(32,3);
// by ID
G=gap.SmallGroup(32,3);
# by ID
G:=PCGroup([5,-2,2,-2,2,-2,20,46,72]);
// Polycyclic
G:=Group<a,b|a^4=b^8=1,a*b=b*a>;
// generators/relations
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