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## G = C22×C18order 72 = 23·32

### Abelian group of type [2,2,18]

Aliases: C22×C18, SmallGroup(72,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C18
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×C18
 Lower central C1 — C22×C18
 Upper central 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 ··· 2G 3A 3B 6A ··· 6N 9A ··· 9F 18A ··· 18AP order 1 2 ··· 2 3 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

72 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C9 C18 kernel C22×C18 C2×C18 C22×C6 C2×C6 C23 C22 # reps 1 7 2 14 6 42

Matrix representation of C22×C18 in GL3(𝔽19) generated by

 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
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

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