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G = C9×Dic6  order 216 = 23·33

Direct product of C9 and Dic6

direct product, metacyclic, supersoluble, monomial

Aliases: C9×Dic6, C36.7S3, C12.1C18, C18.19D6, Dic3.C18, C3⋊(Q8×C9), C4.(S3×C9), (C3×C9)⋊2Q8, C6.29(S3×C6), C2.3(S3×C18), C6.1(C2×C18), (C3×C36).6C2, (C3×Dic6).C3, C12.15(C3×S3), (C3×C12).10C6, C3.4(C3×Dic6), C32.2(C3×Q8), (C3×C18).8C22, (C3×Dic3).2C6, (C9×Dic3).2C2, (C3×C6).18(C2×C6), SmallGroup(216,44)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C9×Dic6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C18 — C9×Dic3 — C9×Dic6
Lower central
C3 — C6 — C9×Dic6
Upper central
C1 — C18 — C36

Generators and relations for C9×Dic6
 G = < a,b,c | a9=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

C1
C2
C3
C3
2C3
C4
3C4
3C4
C6
C6
2C6
C9
C32
2C9
3Q8
C12
Dic3
Dic3
C12
2C12
3C12
3C12
C3×C6
C18
2C18
C3×C9
Dic6
3C3×Q8
C3×C12
C3×Dic3
C3×Dic3
C36
2C36
3C36
3C36
C3×C18
C3×Dic6
3Q8×C9
C3×C36
C9×Dic3
C9×Dic3
C9×Dic6

Smallest permutation representation of C9×Dic6
►On 72 points
Generators in S72
(1 22 70 9 18 66 5 14 62)(2 23 71 10 19 67 6 15 63)(3 24 72 11 20 68 7 16 64)(4 13 61 12 21 69 8 17 65)(25 56 38 29 60 42 33 52 46)(26 57 39 30 49 43 34 53 47)(27 58 40 31 50 44 35 54 48)(28 59 41 32 51 45 36 55 37)

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

C9×Dic6 is a maximal subgroup of
 Dic6⋊D9  C18.D12  C12.D18  C9⋊Dic12  D18.D6  Dic6⋊5D9  Dic18⋊S3  S3×Q8×C9

81 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C6A6B6C6D6E9A···9F9G···9L12A···12H12I12J12K12L18A···18F18G···18L36A···36R36S···36AD
order1233333444666669···99···912···121212121218···1818···1836···3636···36
size1111222266112221···12···22···266661···12···22···26···6

81 irreducible representations

dim111111111222222222222
type++++-+-
imageC1C2C2C3C6C6C9C18C18S3Q8D6C3×S3Dic6C3×Q8S3×C6S3×C9Q8×C9C3×Dic6S3×C18C9×Dic6
kernelC9×Dic6C9×Dic3C3×C36C3×Dic6C3×Dic3C3×C12Dic6Dic3C12C36C3×C9C18C12C9C32C6C4C3C3C2C1
# reps12124261261112222664612

Matrix representation of C9×Dic6 ►in GL2(𝔽37) generated by

160
016
,
140
198
,
279
3410
G:=sub<GL(2,GF(37))| [16,0,0,16],[14,19,0,8],[27,34,9,10] >;
 

C9×Dic6 in GAP, Magma, Sage, TeX 

C_9\times {\rm Dic}_6
 
% in TeX
 
G:=Group("C9xDic6");
 
// GroupNames label
 
G:=SmallGroup(216,44);
 
// by ID
 
G=gap.SmallGroup(216,44);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,122,5189]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C9×Dic6 in TeX

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