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G = C12xD9order 216 = 23·33

Direct product of C12 and D9

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

Aliases: C12xD9, C36:6C6, Dic9:5C6, D18.2C6, C6.20D18, (C3xC36):4C2, C9:4(C2xC12), C6.7(S3xC6), C2.1(C6xD9), C3.1(S3xC12), (C6xD9).2C2, (C3xC6).46D6, C12.11(C3xS3), C18.10(C2xC6), (C3xC12).17S3, (C3xDic9):5C2, C32.4(C4xS3), (C3xC18).14C22, (C3xC9):5(C2xC4), SmallGroup(216,45)

Series: Derived Chief Lower central Upper central

C1C9 — C12xD9
C1C3C9C18C3xC18C6xD9 — C12xD9
C9 — C12xD9
C1C12

Generators and relations for C12xD9
 G = < a,b,c | a12=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 160 in 54 conjugacy classes, 28 normal (24 characteristic)
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C12, D6, C2xC6, D9, C3xS3, C4xS3, C2xC12, D18, S3xC6, C3xD9, C4xD9, S3xC12, C6xD9, C12xD9
9C2
9C2
2C3
9C4
9C22
2C6
3S3
3S3
9C6
9C6
2C9
9C2xC4
2C12
3Dic3
3D6
9C12
9C2xC6
2C18
3C3xS3
3C3xS3
3C4xS3
9C2xC12
2C36
3S3xC6
3C3xDic3
3S3xC12

Smallest permutation representation of C12xD9
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 66 30 5 70 34 9 62 26)(2 67 31 6 71 35 10 63 27)(3 68 32 7 72 36 11 64 28)(4 69 33 8 61 25 12 65 29)(13 45 49 21 41 57 17 37 53)(14 46 50 22 42 58 18 38 54)(15 47 51 23 43 59 19 39 55)(16 48 52 24 44 60 20 40 56)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 36)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 61)(47 62)(48 63)

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

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

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

C12xD9 is a maximal subgroup of   C36.39D6  Dic6:5D9  D12:5D9  D6.D18

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I9A···9I12A12B12C12D12E···12J12K12L12M12N18A···18I36A···36R
order12223333344446666666669···91212121212···121212121218···1836···36
size11991122211991122299992···211112···299992···22···2

72 irreducible representations

dim1111111111222222222222
type++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6D9C3xS3C4xS3D18S3xC6C3xD9C4xD9S3xC12C6xD9C12xD9
kernelC12xD9C3xDic9C3xC36C6xD9C4xD9C3xD9Dic9C36D18D9C3xC12C3xC6C12C12C32C6C6C4C3C3C2C1
# reps11112422281132232664612

Matrix representation of C12xD9 in GL2(F37) generated by

80
08
,
56
60
,
031
60
G:=sub<GL(2,GF(37))| [8,0,0,8],[5,6,6,0],[0,6,31,0] >;

C12xD9 in GAP, Magma, Sage, TeX

C_{12}\times D_9
% in TeX

G:=Group("C12xD9");
// GroupNames label

G:=SmallGroup(216,45);
// by ID

G=gap.SmallGroup(216,45);
# by ID

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

G:=Group<a,b,c|a^12=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C12xD9 in TeX

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