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## G = C22×C8order 32 = 25

### Abelian group of type [2,2,8]

Aliases: C22×C8, SmallGroup(32,36)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8
 Lower central C1 — C22×C8
 Upper central C1 — C22×C8
 Jennings C1 — C2 — C2 — C4 — 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 ··· 2G 4A ··· 4H 8A ··· 8P order 1 2 ··· 2 4 ··· 4 8 ··· 8 size 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 C22×C8 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 16

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

 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
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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