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G = C2xC6xD9  order 216 = 23·33

Direct product of C2xC6 and D9

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2xC6xD9, C62.13S3, C18:3(C2xC6), (C6xC18):6C2, (C3xC9):3C23, (C2xC18):11C6, C6.21(S3xC6), C9:3(C22xC6), (C3xC6).55D6, (C3xC18):3C22, C32.3(C22xS3), C3.1(S3xC2xC6), (C2xC6).16(C3xS3), SmallGroup(216,108)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C9 — C2xC6xD9
Chief series
C1 — C3 — C9 — C3xC9 — C3xD9 — C6xD9 — C2xC6xD9
Lower central
C9 — C2xC6xD9
Upper central
C1 — C2xC6

Generators and relations for C2xC6xD9
 G = < a,b,c,d | a2=b6=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 336 in 106 conjugacy classes, 52 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C9, C9, C32, D6, C2xC6, C2xC6, D9, C18, C18, C3xS3, C3xC6, C22xS3, C22xC6, C3xC9, D18, C2xC18, C2xC18, S3xC6, C62, C3xD9, C3xC18, C22xD9, S3xC2xC6, C6xD9, C6xC18, C2xC6xD9
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2xC6, D9, C3xS3, C22xS3, C22xC6, D18, S3xC6, C3xD9, C22xD9, S3xC2xC6, C6xD9, C2xC6xD9

Smallest permutation representation of C2xC6xD9
►On 72 points
Generators in S72
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)

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

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

C2xC6xD9 is a maximal subgroup of   D18:Dic3

72 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E6A···6F6G···6O6P···6W9A···9I18A···18AA
order12222222333336···66···66···69···918···18
size11119999112221···12···29···92···22···2

72 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6S3D6D9C3xS3D18S3xC6C3xD9C6xD9
kernelC2xC6xD9C6xD9C6xC18C22xD9D18C2xC18C62C3xC6C2xC6C2xC6C6C6C22C2
# reps1612122133296618

Matrix representation of C2xC6xD9 ►in GL3(F19) generated by

100
0180
0018
,
1200
0120
0012
,
100
0170
0139
,
1800
01216
0167
G:=sub<GL(3,GF(19))| [1,0,0,0,18,0,0,0,18],[12,0,0,0,12,0,0,0,12],[1,0,0,0,17,13,0,0,9],[18,0,0,0,12,16,0,16,7] >;
 

C2xC6xD9 in GAP, Magma, Sage, TeX 

C_2\times C_6\times D_9
 
% in TeX
 
G:=Group("C2xC6xD9");
 
// GroupNames label
 
G:=SmallGroup(216,108);
 
// by ID
 
G=gap.SmallGroup(216,108);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,3604,208,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^6=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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