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G = C3×D36order 216 = 23·33

Direct product of C3 and D36

direct product, metacyclic, supersoluble, monomial

Aliases: C3×D36, C365C6, C123D9, D184C6, C6.21D18, C32.3D12, C4⋊(C3×D9), (C3×C9)⋊5D4, C94(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

C1C18 — C3×D36
C1C3C9C18C3×C18C6×D9 — C3×D36
C9C18 — C3×D36
C1C6C12

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

18C2
18C2
2C3
9C22
9C22
2C6
6S3
6S3
18C6
18C6
2C9
9D4
2C12
3D6
3D6
9C2×C6
9C2×C6
2C18
2D9
2D9
6C3×S3
6C3×S3
3D12
9C3×D4
2C36
3S3×C6
3S3×C6
2C3×D9
2C3×D9
3C3×D12

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 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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