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## 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 10)(2 22 32 11)(3 23 25 12)(4 24 26 13)(5 17 27 14)(6 18 28 15)(7 19 29 16)(8 20 30 9)
(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)| (1,21,31,10)(2,22,32,11)(3,23,25,12)(4,24,26,13)(5,17,27,14)(6,18,28,15)(7,19,29,16)(8,20,30,9), (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( (1,21,31,10)(2,22,32,11)(3,23,25,12)(4,24,26,13)(5,17,27,14)(6,18,28,15)(7,19,29,16)(8,20,30,9), (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([(1,21,31,10),(2,22,32,11),(3,23,25,12),(4,24,26,13),(5,17,27,14),(6,18,28,15),(7,19,29,16),(8,20,30,9)], [(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)])

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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