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G = C22×C8order 32 = 25

Abelian group of type [2,2,8]

direct product, p-group, abelian, monomial

Aliases: C22×C8, SmallGroup(32,36)

Series: Derived Chief Lower central Upper central Jennings

C1 — C22×C8
C1C2C4C2×C4C22×C4 — C22×C8
C1 — C22×C8
C1 — C22×C8
C1C2C2C4 — C22×C8

Generators and relations for C22×C8
 G = < a,b,c | a2=b2=c8=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C8
Regular action on 32 points
Generators in S32
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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)| (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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( (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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([(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(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···2G4A···4H8A···8P
order12···24···48···8
size11···11···11···1

32 irreducible representations

dim111111
type+++
imageC1C2C2C4C4C8
kernelC22×C8C2×C8C22×C4C2×C4C23C22
# reps1616216

Matrix representation of C22×C8 in GL3(𝔽17) generated by

100
0160
001
,
100
0160
0016
,
200
020
0013
G:=sub<GL(3,GF(17))| [1,0,0,0,16,0,0,0,1],[1,0,0,0,16,0,0,0,16],[2,0,0,0,2,0,0,0,13] >;

C22×C8 in GAP, Magma, Sage, TeX

C_2^2\times C_8
% in TeX

G:=Group("C2^2xC8");
// GroupNames label

G:=SmallGroup(32,36);
// by ID

G=gap.SmallGroup(32,36);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,40,58]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^8=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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