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G = C2xC22order 44 = 22·11

Abelian group of type [2,22]

direct product, abelian, monomial, 2-elementary

Aliases: C2xC22, SmallGroup(44,4)

Series: Derived Chief Lower central Upper central

C1 — C2xC22
C1C11C22 — C2xC22
C1 — C2xC22
C1 — C2xC22

Generators and relations for C2xC22
 G = < a,b | a2=b22=1, ab=ba >

Subgroups: 10, all normal (4 characteristic)
Quotients: C1, C2, C22, C11, C22, C2xC22

Smallest permutation representation of C2xC22
Regular action on 44 points
Generators in S44
(1 43)(2 44)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)
(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)| (1,43)(2,44)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42), (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( (1,43)(2,44)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42), (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([[(1,43),(2,44),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42)], [(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)]])

C2xC22 is a maximal subgroup of   C11:D4

44 conjugacy classes

class 1 2A2B2C11A···11J22A···22AD
order122211···1122···22
size11111···11···1

44 irreducible representations

dim1111
type++
imageC1C2C11C22
kernelC2xC22C22C22C2
# reps131030

Matrix representation of C2xC22 in GL2(F23) generated by

220
01
,
100
07
G:=sub<GL(2,GF(23))| [22,0,0,1],[10,0,0,7] >;

C2xC22 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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Subgroup lattice of C2xC22 in TeX

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