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G = C3×D36  order 216 = 23·33

Direct product of C3 and D36

direct product, metacyclic, supersoluble, monomial

Aliases: C3×D36, C36⋊5C6, C12⋊3D9, D18⋊4C6, C6.21D18, C32.3D12, C4⋊(C3×D9), (C3×C9)⋊5D4, C9⋊4(C3×D4), (C3×C36)⋊2C2, (C6×D9)⋊3C2, C6.8(S3×C6), C2.4(C6×D9), C12.2(C3×S3), C3.1(C3×D12), (C3×C6).47D6, (C3×C12).13S3, C18.11(C2×C6), (C3×C18).15C22, SmallGroup(216,46)

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 >

C1
C2
18C2
18C2
C3
C3
2C3
C4
9C22
9C22
C6
C6
2C6
6S3
6S3
18C6
18C6
C9
C32
2C9
9D4
C12
C12
2C12
3D6
3D6
9C2×C6
9C2×C6
C18
C3×C6
2C18
2D9
2D9
6C3×S3
6C3×S3
C3×C9
3D12
9C3×D4
C36
C3×C12
D18
D18
2C36
3S3×C6
3S3×C6
C3×C18
2C3×D9
2C3×D9
D36
3C3×D12
C6×D9
C3×C36
C6×D9
C3×D36

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  D36⋊5S3  C36⋊D6  C3×D4×D9

63 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C6D6E6F6G6H6I9A···9I12A···12H18A···18I36A···36R
order12223333346666666669···912···1218···1836···36
size11181811222211222181818182···22···22···22···2

63 irreducible representations

dim11111122222222222222
type++++++++++
imageC1C2C2C3C6C6S3D4D6D9C3×S3C3×D4D12D18S3×C6C3×D9D36C3×D12C6×D9C3×D36
kernelC3×D36C3×C36C6×D9D36C36D18C3×C12C3×C9C3×C6C12C12C9C32C6C6C4C3C3C2C1
# reps112224111322232664612

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

260
026
,
130
120
,
3132
76
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
 

Export

Subgroup lattice of C3×D36 in TeX

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