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## G = C3×D36order 216 = 23·33

### Direct product of C3 and D36

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×D36
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C3×D36
 Lower central C9 — C18 — C3×D36
 Upper central C1 — C6 — C12

Generators and relations for C3×D36
G = < a,b,c | a3=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C3×D36
On 72 points
Generators in S72
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)

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

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

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

C3×D36 is a maximal subgroup of
D36.S3  C3⋊D72  D36⋊S3  Dic6⋊D9  D18.D6  D365S3  C36⋊D6  C3×D4×D9

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 D9 C3×S3 C3×D4 D12 D18 S3×C6 C3×D9 D36 C3×D12 C6×D9 C3×D36 kernel C3×D36 C3×C36 C6×D9 D36 C36 D18 C3×C12 C3×C9 C3×C6 C12 C12 C9 C32 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 1 3 2 2 2 3 2 6 6 4 6 12

Matrix representation of C3×D36 in GL2(𝔽37) generated by

 26 0 0 26
,
 13 0 1 20
,
 31 32 7 6
G:=sub<GL(2,GF(37))| [26,0,0,26],[13,1,0,20],[31,7,32,6] >;

C3×D36 in GAP, Magma, Sage, TeX

C_3\times D_{36}
% in TeX

G:=Group("C3xD36");
// GroupNames label

G:=SmallGroup(216,46);
// by ID

G=gap.SmallGroup(216,46);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,3604,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^3=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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