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G = C22×C22order 88 = 23·11

Abelian group of type [2,2,22]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C22, SmallGroup(88,12)

Series: Derived Chief Lower central Upper central

C1 — C22×C22
C1C11C22C2×C22 — C22×C22
C1 — C22×C22
C1 — C22×C22

Generators and relations for C22×C22
 G = < a,b,c | a2=b2=c22=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C22
Regular action on 88 points
Generators in S88
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)
(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)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 67)(61 68)(62 69)(63 70)(64 71)(65 72)(66 73)
(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)(45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66)(67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88)

G:=sub<Sym(88)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75), (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)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,67)(61,68)(62,69)(63,70)(64,71)(65,72)(66,73), (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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)>;

G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75), (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)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,67)(61,68)(62,69)(63,70)(64,71)(65,72)(66,73), (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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88) );

G=PermutationGroup([(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,67),(37,68),(38,69),(39,70),(40,71),(41,72),(42,73),(43,74),(44,75)], [(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),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,85),(57,86),(58,87),(59,88),(60,67),(61,68),(62,69),(63,70),(64,71),(65,72),(66,73)], [(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),(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66),(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)])

C22×C22 is a maximal subgroup of   C23.D11

88 conjugacy classes

class 1 2A···2G11A···11J22A···22BR
order12···211···1122···22
size11···11···11···1

88 irreducible representations

dim1111
type++
imageC1C2C11C22
kernelC22×C22C2×C22C23C22
# reps171070

Matrix representation of C22×C22 in GL3(𝔽23) generated by

2200
0220
0022
,
100
0220
001
,
2000
0210
0018
G:=sub<GL(3,GF(23))| [22,0,0,0,22,0,0,0,22],[1,0,0,0,22,0,0,0,1],[20,0,0,0,21,0,0,0,18] >;

C22×C22 in GAP, Magma, Sage, TeX

C_2^2\times C_{22}
% in TeX

G:=Group("C2^2xC22");
// GroupNames label

G:=SmallGroup(88,12);
// by ID

G=gap.SmallGroup(88,12);
# by ID

G:=PCGroup([4,-2,-2,-2,-11]);
// Polycyclic

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

Export

Subgroup lattice of C22×C22 in TeX

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