Copied to
clipboard

## G = C4×C8order 32 = 25

### Abelian group of type [4,8]

Aliases: C4×C8, SmallGroup(32,3)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4×C8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8
 Lower central C1 — C4×C8
 Upper central C1 — C4×C8
 Jennings C1 — C2 — C2 — C2×C4 — 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  C82C8  C81C8  C165C4  C4⋊C16  C8.C8  C82M4(2)  C42.12C4  C42.7C22  C86D4  C8○D8  C84Q8  C4.4D8  C4.SD16  C42.78C22  C85D4  C84D4  C4⋊Q16  C8.12D4  C83Q8  C8.5Q8  C82Q8
C4×C8 is a maximal quotient of
C8⋊C8  C22.7C42  C165C4

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

Export

׿
×
𝔽