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G = C9×D12order 216 = 23·33

Direct product of C9 and D12

direct product, metacyclic, supersoluble, monomial

Aliases: C9×D12, C363S3, C121C18, D61C18, C18.21D6, C4⋊(S3×C9), (C3×C9)⋊4D4, C31(D4×C9), (C3×C36)⋊6C2, (C3×D12).C3, (S3×C18)⋊1C2, (S3×C6).1C6, C2.4(S3×C18), C6.31(S3×C6), C6.3(C2×C18), C3.4(C3×D12), C12.16(C3×S3), (C3×C12).11C6, C32.2(C3×D4), (C3×C18).10C22, (C3×C6).20(C2×C6), SmallGroup(216,48)

Series: Derived Chief Lower central Upper central

C1C6 — C9×D12
C1C3C32C3×C6C3×C18S3×C18 — C9×D12
C3C6 — C9×D12
C1C18C36

Generators and relations for C9×D12
 G = < a,b,c | a9=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

6C2
6C2
2C3
3C22
3C22
2S3
2C6
2S3
6C6
6C6
2C9
3D4
2C12
3C2×C6
3C2×C6
2C18
2C3×S3
2C3×S3
6C18
6C18
3C3×D4
2C36
3C2×C18
3C2×C18
2S3×C9
2S3×C9
3D4×C9

Smallest permutation representation of C9×D12
On 72 points
Generators in S72
(1 61 47 5 65 39 9 69 43)(2 62 48 6 66 40 10 70 44)(3 63 37 7 67 41 11 71 45)(4 64 38 8 68 42 12 72 46)(13 55 31 21 51 27 17 59 35)(14 56 32 22 52 28 18 60 36)(15 57 33 23 53 29 19 49 25)(16 58 34 24 54 30 20 50 26)
(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)(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 60)(12 59)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 72)(36 71)

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

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

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

C9×D12 is a maximal subgroup of
D36⋊S3  C9⋊D24  D12.D9  C36.D6  D125D9  D12⋊D9  C36⋊D6  S3×D4×C9

81 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C6D6E6F6G6H6I9A···9F9G···9L12A···12H18A···18F18G···18L18M···18X36A···36R
order12223333346666666669···99···912···1218···1818···1818···1836···36
size11661122221122266661···12···22···21···12···26···62···2

81 irreducible representations

dim111111111222222222222
type+++++++
imageC1C2C2C3C6C6C9C18C18S3D4D6C3×S3D12C3×D4S3×C6S3×C9D4×C9C3×D12S3×C18C9×D12
kernelC9×D12C3×C36S3×C18C3×D12C3×C12S3×C6D12C12D6C36C3×C9C18C12C9C32C6C4C3C3C2C1
# reps11222466121112222664612

Matrix representation of C9×D12 in GL2(𝔽37) generated by

70
07
,
230
029
,
030
210
G:=sub<GL(2,GF(37))| [7,0,0,7],[23,0,0,29],[0,21,30,0] >;

C9×D12 in GAP, Magma, Sage, TeX

C_9\times D_{12}
% in TeX

G:=Group("C9xD12");
// GroupNames label

G:=SmallGroup(216,48);
// by ID

G=gap.SmallGroup(216,48);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,122,5189]);
// Polycyclic

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

Export

Subgroup lattice of C9×D12 in TeX

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