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G = C12×D9order 216 = 23·33

Direct product of C12 and D9

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

Aliases: C12×D9, C366C6, Dic95C6, D18.2C6, C6.20D18, (C3×C36)⋊4C2, C94(C2×C12), C6.7(S3×C6), C2.1(C6×D9), C3.1(S3×C12), (C6×D9).2C2, (C3×C6).46D6, C12.11(C3×S3), C18.10(C2×C6), (C3×C12).17S3, (C3×Dic9)⋊5C2, C32.4(C4×S3), (C3×C18).14C22, (C3×C9)⋊5(C2×C4), SmallGroup(216,45)

Series: Derived Chief Lower central Upper central

C1C9 — C12×D9
C1C3C9C18C3×C18C6×D9 — C12×D9
C9 — C12×D9
C1C12

Generators and relations for C12×D9
 G = < a,b,c | a12=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
9C2
2C3
9C4
9C22
2C6
3S3
3S3
9C6
9C6
2C9
9C2×C4
2C12
3Dic3
3D6
9C12
9C2×C6
2C18
3C3×S3
3C3×S3
3C4×S3
9C2×C12
2C36
3S3×C6
3C3×Dic3
3S3×C12

Smallest permutation representation of C12×D9
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 58 64 5 50 68 9 54 72)(2 59 65 6 51 69 10 55 61)(3 60 66 7 52 70 11 56 62)(4 49 67 8 53 71 12 57 63)(13 25 41 21 33 37 17 29 45)(14 26 42 22 34 38 18 30 46)(15 27 43 23 35 39 19 31 47)(16 28 44 24 36 40 20 32 48)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 71)(26 72)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)

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

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

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

C12×D9 is a maximal subgroup of   C36.39D6  Dic65D9  D125D9  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++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6D9C3×S3C4×S3D18S3×C6C3×D9C4×D9S3×C12C6×D9C12×D9
kernelC12×D9C3×Dic9C3×C36C6×D9C4×D9C3×D9Dic9C36D18D9C3×C12C3×C6C12C12C32C6C6C4C3C3C2C1
# reps11112422281132232664612

Matrix representation of C12×D9 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] >;

C12×D9 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 C12×D9 in TeX

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