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G = C22xC18order 72 = 23·32

Abelian group of type [2,2,18]

direct product, abelian, monomial, 2-elementary

Aliases: C22xC18, SmallGroup(72,18)

Series: Derived Chief Lower central Upper central

C1 — C22xC18
C1C3C9C18C2xC18 — C22xC18
C1 — C22xC18
C1 — C22xC18

Generators and relations for C22xC18
 G = < a,b,c | a2=b2=c18=1, ab=ba, ac=ca, bc=cb >

Subgroups: 48, all normal (6 characteristic)
Quotients: C1, C2, C3, C22, C6, C23, C9, C2xC6, C18, C22xC6, C2xC18, C22xC18

Smallest permutation representation of C22xC18
Regular action on 72 points
Generators in S72
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(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)

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

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

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

C22xC18 is a maximal subgroup of   C18.D4

72 conjugacy classes

class 1 2A···2G3A3B6A···6N9A···9F18A···18AP
order12···2336···69···918···18
size11···1111···11···11···1

72 irreducible representations

dim111111
type++
imageC1C2C3C6C9C18
kernelC22xC18C2xC18C22xC6C2xC6C23C22
# reps17214642

Matrix representation of C22xC18 in GL3(F19) generated by

100
0180
001
,
1800
0180
0018
,
300
020
0017
G:=sub<GL(3,GF(19))| [1,0,0,0,18,0,0,0,1],[18,0,0,0,18,0,0,0,18],[3,0,0,0,2,0,0,0,17] >;

C22xC18 in GAP, Magma, Sage, TeX

C_2^2\times C_{18}
% in TeX

G:=Group("C2^2xC18");
// GroupNames label

G:=SmallGroup(72,18);
// by ID

G=gap.SmallGroup(72,18);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,78]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^18=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of C22xC18 in TeX

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