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G = S3×C36order 216 = 23·33

Direct product of C36 and S3

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: S3×C36, D6.C18, C122C18, C18.20D6, Dic32C18, (C3×C36)⋊1C2, C31(C2×C36), (S3×C12).C3, (C3×S3).C12, (S3×C6).4C6, C6.30(S3×C6), C3.4(S3×C12), C2.1(S3×C18), C6.2(C2×C18), C36(C3×Dic3), C36(C9×Dic3), (S3×C18).2C2, C12.19(C3×S3), (C3×C12).14C6, (C9×Dic3)⋊5C2, C32.2(C2×C12), (C3×C18).9C22, (C3×Dic3).5C6, (C3×C9)⋊4(C2×C4), (C3×C6).19(C2×C6), SmallGroup(216,47)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C36
C1C3C32C3×C6C3×C18S3×C18 — S3×C36
C3 — S3×C36
C1C36

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

3C2
3C2
2C3
3C4
3C22
2C6
3C6
3C6
2C9
3C2×C4
2C12
3C12
3C2×C6
2C18
3C18
3C18
3C2×C12
2C36
3C2×C18
3C36
3C2×C36

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

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

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

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

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

108 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I9A···9F9G···9L12A12B12C12D12E···12J12K12L12M12N18A···18F18G···18L18M···18X36A···36L36M···36X36Y···36AJ
order12223333344446666666669···99···91212121212···121212121218···1818···1818···1836···3636···3636···36
size11331122211331122233331···12···211112···233331···12···23···31···12···23···3

108 irreducible representations

dim111111111111111222222222
type++++++
imageC1C2C2C2C3C4C6C6C6C9C12C18C18C18C36S3D6C3×S3C4×S3S3×C6S3×C9S3×C12S3×C18S3×C36
kernelS3×C36C9×Dic3C3×C36S3×C18S3×C12S3×C9C3×Dic3C3×C12S3×C6C4×S3C3×S3Dic3C12D6S3C36C18C12C9C6C4C3C2C1
# reps11112422268666241122264612

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

190
019
,
1029
026
,
2334
2814
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

Export

Subgroup lattice of S3×C36 in TeX

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