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G = C6×Dic9order 216 = 23·33

Direct product of C6 and Dic9

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

Aliases: C6×Dic9, C183C12, C6.22D18, C62.9S3, C95(C2×C12), (C3×C18)⋊2C4, (C2×C6).6D9, C2.2(C6×D9), C22.(C3×D9), C6.15(S3×C6), (C2×C18).5C6, (C6×C18).2C2, (C3×C6).48D6, C18.12(C2×C6), C6.4(C3×Dic3), C3.1(C6×Dic3), (C3×C6).8Dic3, (C3×C18).16C22, C32.3(C2×Dic3), (C3×C9)⋊8(C2×C4), (C2×C6).10(C3×S3), SmallGroup(216,55)

Series: Derived Chief Lower central Upper central

C1C9 — C6×Dic9
C1C3C9C18C3×C18C3×Dic9 — C6×Dic9
C9 — C6×Dic9
C1C2×C6

Generators and relations for C6×Dic9
 G = < a,b,c | a6=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 128 in 58 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C9, C9, C32, Dic3 [×2], C12 [×2], C2×C6 [×2], C2×C6, C18, C18 [×2], C18 [×3], C3×C6, C3×C6 [×2], C2×Dic3, C2×C12, C3×C9, Dic9 [×2], C2×C18, C2×C18, C3×Dic3 [×2], C62, C3×C18, C3×C18 [×2], C2×Dic9, C6×Dic3, C3×Dic9 [×2], C6×C18, C6×Dic9
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, D9, C3×S3, C2×Dic3, C2×C12, Dic9 [×2], D18, C3×Dic3 [×2], S3×C6, C3×D9, C2×Dic9, C6×Dic3, C3×Dic9 [×2], C6×D9, C6×Dic9

Smallest permutation representation of C6×Dic9
On 72 points
Generators in S72
(1 34 13 28 7 22)(2 35 14 29 8 23)(3 36 15 30 9 24)(4 19 16 31 10 25)(5 20 17 32 11 26)(6 21 18 33 12 27)(37 67 43 55 49 61)(38 68 44 56 50 62)(39 69 45 57 51 63)(40 70 46 58 52 64)(41 71 47 59 53 65)(42 72 48 60 54 66)
(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 39 10 48)(2 38 11 47)(3 37 12 46)(4 54 13 45)(5 53 14 44)(6 52 15 43)(7 51 16 42)(8 50 17 41)(9 49 18 40)(19 66 28 57)(20 65 29 56)(21 64 30 55)(22 63 31 72)(23 62 32 71)(24 61 33 70)(25 60 34 69)(26 59 35 68)(27 58 36 67)

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

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

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

C6×Dic9 is a maximal subgroup of   Dic9⋊Dic3  C18.Dic6  Dic3⋊Dic9  C6.18D36  D6⋊Dic9  Dic3.D18  D9×C2×C12

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A···6F6G···6O9A···9I12A···12H18A···18AA
order12223333344446···66···69···912···1218···18
size11111122299991···12···22···29···92···2

72 irreducible representations

dim11111111222222222222
type++++-++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6D9C3×S3Dic9D18C3×Dic3S3×C6C3×D9C3×Dic9C6×D9
kernelC6×Dic9C3×Dic9C6×C18C2×Dic9C3×C18Dic9C2×C18C18C62C3×C6C3×C6C2×C6C2×C6C6C6C6C6C22C2C2
# reps121244281213263426126

Matrix representation of C6×Dic9 in GL3(𝔽37) generated by

2700
0270
0027
,
100
0250
003
,
3600
001
0360
G:=sub<GL(3,GF(37))| [27,0,0,0,27,0,0,0,27],[1,0,0,0,25,0,0,0,3],[36,0,0,0,0,36,0,1,0] >;

C6×Dic9 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_9
% in TeX

G:=Group("C6xDic9");
// GroupNames label

G:=SmallGroup(216,55);
// by ID

G=gap.SmallGroup(216,55);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,3604,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^6=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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