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## G = S3×C36order 216 = 23·33

### Direct product of C36 and S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C36
G = < a,b,c | a36=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C36
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 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 37)(35 38)(36 39)

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

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

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

S3×C36 is a maximal subgroup of   D6.Dic9  D6.D18  D365S3  Dic9.D6

108 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 ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 18A ··· 18F 18G ··· 18L 18M ··· 18X 36A ··· 36L 36M ··· 36X 36Y ··· 36AJ 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 9 ··· 9 12 12 12 12 12 ··· 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 3 3 1 1 2 2 2 1 1 3 3 1 1 2 2 2 3 3 3 3 1 ··· 1 2 ··· 2 1 1 1 1 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C9 C12 C18 C18 C18 C36 S3 D6 C3×S3 C4×S3 S3×C6 S3×C9 S3×C12 S3×C18 S3×C36 kernel S3×C36 C9×Dic3 C3×C36 S3×C18 S3×C12 S3×C9 C3×Dic3 C3×C12 S3×C6 C4×S3 C3×S3 Dic3 C12 D6 S3 C36 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 6 8 6 6 6 24 1 1 2 2 2 6 4 6 12

Matrix representation of S3×C36 in GL2(𝔽37) generated by

 19 0 0 19
,
 10 29 0 26
,
 23 34 28 14
G:=sub<GL(2,GF(37))| [19,0,0,19],[10,0,29,26],[23,28,34,14] >;

S3×C36 in GAP, Magma, Sage, TeX

S_3\times C_{36}
% in TeX

G:=Group("S3xC36");
// GroupNames label

G:=SmallGroup(216,47);
// by ID

G=gap.SmallGroup(216,47);
# by ID

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

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

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