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## G = C12×D9order 216 = 23·33

### Direct product of C12 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C12×D9
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C12×D9
 Lower central C9 — C12×D9
 Upper central C1 — C12

Generators and relations for C12×D9
G = < a,b,c | a12=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C12×D9
On 72 points
Generators in S72
(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 64 5 50 68 9 54 72)(2 59 65 6 51 69 10 55 61)(3 60 66 7 52 70 11 56 62)(4 49 67 8 53 71 12 57 63)(13 25 41 21 33 37 17 29 45)(14 26 42 22 34 38 18 30 46)(15 27 43 23 35 39 19 31 47)(16 28 44 24 36 40 20 32 48)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 71)(26 72)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)

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

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

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

C12×D9 is a maximal subgroup of   C36.39D6  Dic65D9  D125D9  D6.D18

72 conjugacy classes

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

72 irreducible representations

 dim 1 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 C2 C3 C4 C6 C6 C6 C12 S3 D6 D9 C3×S3 C4×S3 D18 S3×C6 C3×D9 C4×D9 S3×C12 C6×D9 C12×D9 kernel C12×D9 C3×Dic9 C3×C36 C6×D9 C4×D9 C3×D9 Dic9 C36 D18 D9 C3×C12 C3×C6 C12 C12 C32 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 3 2 2 3 2 6 6 4 6 12

Matrix representation of C12×D9 in GL2(𝔽37) generated by

 8 0 0 8
,
 5 6 6 0
,
 0 31 6 0
G:=sub<GL(2,GF(37))| [8,0,0,8],[5,6,6,0],[0,6,31,0] >;

C12×D9 in GAP, Magma, Sage, TeX

C_{12}\times D_9
% in TeX

G:=Group("C12xD9");
// GroupNames label

G:=SmallGroup(216,45);
// by ID

G=gap.SmallGroup(216,45);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,3604,208,5189]);
// Polycyclic

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

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