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## G = C2×C22order 44 = 22·11

### Abelian group of type [2,22]

Aliases: C2×C22, SmallGroup(44,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22
 Chief series C1 — C11 — C22 — C2×C22
 Lower central C1 — C2×C22
 Upper central C1 — C2×C22

Generators and relations for C2×C22
G = < a,b | a2=b22=1, ab=ba >

Smallest permutation representation of C2×C22
Regular action on 44 points
Generators in S44
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)
(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 33 34 35 36 37 38 39 40 41 42 43 44)

G:=sub<Sym(44)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)>;

G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44) );

G=PermutationGroup([(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)])

C2×C22 is a maximal subgroup of   C11⋊D4

44 conjugacy classes

 class 1 2A 2B 2C 11A ··· 11J 22A ··· 22AD order 1 2 2 2 11 ··· 11 22 ··· 22 size 1 1 1 1 1 ··· 1 1 ··· 1

44 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C11 C22 kernel C2×C22 C22 C22 C2 # reps 1 3 10 30

Matrix representation of C2×C22 in GL2(𝔽23) generated by

 22 0 0 1
,
 10 0 0 7
G:=sub<GL(2,GF(23))| [22,0,0,1],[10,0,0,7] >;

C2×C22 in GAP, Magma, Sage, TeX

C_2\times C_{22}
% in TeX

G:=Group("C2xC22");
// GroupNames label

G:=SmallGroup(44,4);
// by ID

G=gap.SmallGroup(44,4);
# by ID

G:=PCGroup([3,-2,-2,-11]);
// Polycyclic

G:=Group<a,b|a^2=b^22=1,a*b=b*a>;
// generators/relations

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