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G = C22×C18order 72 = 23·32

Abelian group of type [2,2,18]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C18, SmallGroup(72,18)

Series: Derived Chief Lower central Upper central

C1 — C22×C18
C1C3C9C18C2×C18 — C22×C18
C1 — C22×C18
C1 — C22×C18

Generators and relations for C22×C18
 G = < a,b,c | a2=b2=c18=1, ab=ba, ac=ca, bc=cb >


Smallest permutation representation of C22×C18
Regular action on 72 points
Generators in S72
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 37)(17 38)(18 39)(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 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 19)(17 20)(18 21)(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)| (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,37)(17,38)(18,39)(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), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,19)(17,20)(18,21)(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( (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,37)(17,38)(18,39)(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), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,19)(17,20)(18,21)(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([(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,37),(17,38),(18,39),(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)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,19),(17,20),(18,21),(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)])

C22×C18 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
kernelC22×C18C2×C18C22×C6C2×C6C23C22
# reps17214642

Matrix representation of C22×C18 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] >;

C22×C18 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 C22×C18 in TeX

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