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G = Dic9⋊C6  order 216 = 23·33

The semidirect product of Dic9 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: Dic9⋊C6, D18⋊2C6, C62.3S3, 3- 1+2⋊2D4, C9⋊C12⋊C2, C9⋊D4⋊C3, C9⋊2(C3×D4), (C2×C18)⋊3C6, C6.20(S3×C6), C18.5(C2×C6), (C3×C6).15D6, C22⋊3(C9⋊C6), C32.(C3⋊D4), (C22×3- 1+2)⋊1C2, (C2×3- 1+2).5C22, (C2×C9⋊C6)⋊2C2, C2.5(C2×C9⋊C6), C3.3(C3×C3⋊D4), (C2×C6).15(C3×S3), SmallGroup(216,62)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — Dic9⋊C6
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — Dic9⋊C6
Lower central
C9 — C18 — Dic9⋊C6
Upper central
C1 — C2 — C22

Generators and relations for Dic9⋊C6
 G = < a,b,c | a18=c6=1, b2=a9, bab-1=a-1, cac-1=a7, cbc-1=a9b >

C1
C2
2C2
18C2
C3
3C3
C22
9C22
9C4
C6
2C6
3C6
6C6
6S3
18C6
C32
C9
2C9
9D4
C2×C6
3Dic3
3C2×C6
3D6
9C12
9C2×C6
C18
C3×C6
2C3×C6
2C18
2C18
2C18
2D9
2C18
6C3×S3
3- 1+2
3C3⋊D4
9C3×D4
C62
C2×C18
D18
Dic9
2C2×C18
3S3×C6
3C3×Dic3
C2×3- 1+2
2C9⋊C6
2C2×3- 1+2
C9⋊D4
3C3×C3⋊D4
C2×C9⋊C6
C9⋊C12
C22×3- 1+2
Dic9⋊C6

Smallest permutation representation of Dic9⋊C6
►On 36 points
Generators in S36
(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)

(1 20 10 29)(2 19 11 28)(3 36 12 27)(4 35 13 26)(5 34 14 25)(6 33 15 24)(7 32 16 23)(8 31 17 22)(9 30 18 21)

(1 20)(2 33 8 21 14 27)(3 28 15 22 9 34)(4 23)(5 36 11 24 17 30)(6 31 18 25 12 19)(7 26)(10 29)(13 32)(16 35)
 
G:=sub<Sym(36)| (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), (1,20,10,29)(2,19,11,28)(3,36,12,27)(4,35,13,26)(5,34,14,25)(6,33,15,24)(7,32,16,23)(8,31,17,22)(9,30,18,21), (1,20)(2,33,8,21,14,27)(3,28,15,22,9,34)(4,23)(5,36,11,24,17,30)(6,31,18,25,12,19)(7,26)(10,29)(13,32)(16,35)>;
 
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), (1,20,10,29)(2,19,11,28)(3,36,12,27)(4,35,13,26)(5,34,14,25)(6,33,15,24)(7,32,16,23)(8,31,17,22)(9,30,18,21), (1,20)(2,33,8,21,14,27)(3,28,15,22,9,34)(4,23)(5,36,11,24,17,30)(6,31,18,25,12,19)(7,26)(10,29)(13,32)(16,35) );
 
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)], [(1,20,10,29),(2,19,11,28),(3,36,12,27),(4,35,13,26),(5,34,14,25),(6,33,15,24),(7,32,16,23),(8,31,17,22),(9,30,18,21)], [(1,20),(2,33,8,21,14,27),(3,28,15,22,9,34),(4,23),(5,36,11,24,17,30),(6,31,18,25,12,19),(7,26),(10,29),(13,32),(16,35)]])
 

Dic9⋊C6 is a maximal subgroup of   D36⋊6C6  D4×C9⋊C6  Dic18⋊2C6
Dic9⋊C6 is a maximal quotient of   Dic9⋊C12  D18⋊C12  Dic18⋊C6  D36⋊C6  Dic18.C6  D36.C6  C62.27D6

31 conjugacy classes

class 1 2A2B2C3A3B3C 4 6A6B6C6D6E6F6G6H6I9A9B9C12A12B18A···18I
order12223334666666666999121218···18
size11218233182223366181866618186···6

31 irreducible representations

dim1111111122222222666
type+++++++++
imageC1C2C2C2C3C6C6C6S3D4D6C3×S3C3×D4C3⋊D4S3×C6C3×C3⋊D4C9⋊C6C2×C9⋊C6Dic9⋊C6
kernelDic9⋊C6C9⋊C12C2×C9⋊C6C22×3- 1+2C9⋊D4Dic9D18C2×C18C623- 1+2C3×C6C2×C6C9C32C6C3C22C2C1
# reps1111222211122224112

Matrix representation of Dic9⋊C6 ►in GL6(𝔽37)

001100
0036000
000011
0000360
0360000
110000
,
30230000
3070000
0000147
00003023
0014700
00302300
,
7140000
23300000
00233000
0073000
0000730
0000714

G:=sub<GL(6,GF(37))| [0,0,0,0,0,1,0,0,0,0,36,1,1,36,0,0,0,0,1,0,0,0,0,0,0,0,1,36,0,0,0,0,1,0,0,0],[30,30,0,0,0,0,23,7,0,0,0,0,0,0,0,0,14,30,0,0,0,0,7,23,0,0,14,30,0,0,0,0,7,23,0,0],[7,23,0,0,0,0,14,30,0,0,0,0,0,0,23,7,0,0,0,0,30,30,0,0,0,0,0,0,7,7,0,0,0,0,30,14] >;
 

Dic9⋊C6 in GAP, Magma, Sage, TeX 

{\rm Dic}_9\rtimes C_6
 
% in TeX
 
G:=Group("Dic9:C6");
 
// GroupNames label
 
G:=SmallGroup(216,62);
 
// by ID
 
G=gap.SmallGroup(216,62);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,3604,736,208,5189]);
 
// Polycyclic
 
G:=Group<a,b,c|a^18=c^6=1,b^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
 
// generators/relations
 

Export

Subgroup lattice of Dic9⋊C6 in TeX

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