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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 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(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 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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)| (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(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)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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( (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(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)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (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([(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(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),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(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)])

C22×C8 is a maximal subgroup of
C22.7C42  C22.4Q16  C4.C42  C22⋊C16  C82M4(2)  (C22×C8)⋊C2  C23.24D4  C42.6C22  C23.25D4  C89D4  C88D4  C87D4  C8.18D4
C22×C8 is a maximal quotient of
C42.12C4  D4○C16

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

Export

Subgroup lattice of C22×C8 in TeX

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