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## G = C9×D12order 216 = 23·33

### Direct product of C9 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — C9×D12
 Lower central C3 — C6 — C9×D12
 Upper central C1 — C18 — C36

Generators and relations for C9×D12
G = < a,b,c | a9=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C9×D12
On 72 points
Generators in S72
(1 61 47 5 65 39 9 69 43)(2 62 48 6 66 40 10 70 44)(3 63 37 7 67 41 11 71 45)(4 64 38 8 68 42 12 72 46)(13 55 31 21 51 27 17 59 35)(14 56 32 22 52 28 18 60 36)(15 57 33 23 53 29 19 49 25)(16 58 34 24 54 30 20 50 26)
(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 58)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 60)(12 59)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 72)(36 71)

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

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

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

C9×D12 is a maximal subgroup of
D36⋊S3  C9⋊D24  D12.D9  C36.D6  D125D9  D12⋊D9  C36⋊D6  S3×D4×C9

81 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9F 9G ··· 9L 12A ··· 12H 18A ··· 18F 18G ··· 18L 18M ··· 18X 36A ··· 36R order 1 2 2 2 3 3 3 3 3 4 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 6 6 1 1 2 2 2 2 1 1 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 D4 D6 C3×S3 D12 C3×D4 S3×C6 S3×C9 D4×C9 C3×D12 S3×C18 C9×D12 kernel C9×D12 C3×C36 S3×C18 C3×D12 C3×C12 S3×C6 D12 C12 D6 C36 C3×C9 C18 C12 C9 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 6 6 12 1 1 1 2 2 2 2 6 6 4 6 12

Matrix representation of C9×D12 in GL2(𝔽37) generated by

 7 0 0 7
,
 23 0 0 29
,
 0 30 21 0
G:=sub<GL(2,GF(37))| [7,0,0,7],[23,0,0,29],[0,21,30,0] >;

C9×D12 in GAP, Magma, Sage, TeX

C_9\times D_{12}
% in TeX

G:=Group("C9xD12");
// GroupNames label

G:=SmallGroup(216,48);
// by ID

G=gap.SmallGroup(216,48);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,122,5189]);
// Polycyclic

G:=Group<a,b,c|a^9=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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