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G = C4×C8order 32 = 25

Abelian group of type [4,8]

direct product, p-group, abelian, monomial

Aliases: C4×C8, SmallGroup(32,3)

Series: Derived Chief Lower central Upper central Jennings

C1 — C4×C8
C1C2C22C2×C4C42 — C4×C8
C1 — C4×C8
C1 — C4×C8
C1C2C2C2×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 2A2B2C4A···4L8A···8P
order12224···48···8
size11111···11···1

32 irreducible representations

dim111111
type+++
imageC1C2C2C4C4C8
kernelC4×C8C42C2×C8C8C2×C4C4
# reps1128416

Matrix representation of C4×C8 in GL2(𝔽17) generated by

40
04
,
20
04
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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