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## G = C2×C6×D9order 216 = 23·33

### Direct product of C2×C6 and D9

Aliases: C2×C6×D9, C62.13S3, C183(C2×C6), (C6×C18)⋊6C2, (C3×C9)⋊3C23, (C2×C18)⋊11C6, C6.21(S3×C6), C93(C22×C6), (C3×C6).55D6, (C3×C18)⋊3C22, C32.3(C22×S3), C3.1(S3×C2×C6), (C2×C6).16(C3×S3), SmallGroup(216,108)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C6×D9
 Chief series C1 — C3 — C9 — C3×C9 — C3×D9 — C6×D9 — C2×C6×D9
 Lower central C9 — C2×C6×D9
 Upper central C1 — C2×C6

Generators and relations for C2×C6×D9
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, C2×C6, C2×C6, D9, C18, C18, C3×S3, C3×C6, C22×S3, C22×C6, C3×C9, D18, C2×C18, C2×C18, S3×C6, C62, C3×D9, C3×C18, C22×D9, S3×C2×C6, C6×D9, C6×C18, C2×C6×D9
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, D9, C3×S3, C22×S3, C22×C6, D18, S3×C6, C3×D9, C22×D9, S3×C2×C6, C6×D9, C2×C6×D9

Smallest permutation representation of C2×C6×D9
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)]])

C2×C6×D9 is a maximal subgroup of   D18⋊Dic3

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 6A ··· 6F 6G ··· 6O 6P ··· 6W 9A ··· 9I 18A ··· 18AA order 1 2 2 2 2 2 2 2 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 9 9 9 9 1 1 2 2 2 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 D9 C3×S3 D18 S3×C6 C3×D9 C6×D9 kernel C2×C6×D9 C6×D9 C6×C18 C22×D9 D18 C2×C18 C62 C3×C6 C2×C6 C2×C6 C6 C6 C22 C2 # reps 1 6 1 2 12 2 1 3 3 2 9 6 6 18

Matrix representation of C2×C6×D9 in GL3(𝔽19) generated by

 1 0 0 0 18 0 0 0 18
,
 12 0 0 0 12 0 0 0 12
,
 1 0 0 0 17 0 0 13 9
,
 18 0 0 0 12 16 0 16 7
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] >;

C2×C6×D9 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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