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G = C6×C9⋊C6  order 324 = 22·34

Direct product of C6 and C9⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: C6×C9⋊C6, C9⋊C62, D18⋊C32, C33.5D6, C18⋊(C3×C6), (C6×D9)⋊C3, D9⋊(C3×C6), (C3×C18)⋊4C6, (C3×D9)⋊2C6, C6.6(S3×C32), C32.18(S3×C6), (C32×C6).13S3, (C6×3- 1+2)⋊1C2, (C2×3- 1+2)⋊3C6, 3- 1+2⋊3(C2×C6), (C3×3- 1+2)⋊2C22, C3.3(S3×C3×C6), (C3×C9)⋊5(C2×C6), (C3×C6).11(C3×S3), SmallGroup(324,140)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C6×C9⋊C6
 G = < a,b,c | a6=b9=c6=1, ab=ba, ac=ca, cbc-1=b2 >

Subgroups: 334 in 110 conjugacy classes, 46 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C9, C32, C32, C32, D6, C2×C6, D9, C18, C18, C3×S3, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, D18, S3×C6, C62, C3×D9, C9⋊C6, C3×C18, C3×C18, C2×3- 1+2, C2×3- 1+2, S3×C32, C32×C6, C3×3- 1+2, C6×D9, C2×C9⋊C6, S3×C3×C6, C3×C9⋊C6, C6×3- 1+2, C6×C9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, C3×S3, C3×C6, S3×C6, C62, C9⋊C6, S3×C32, C2×C9⋊C6, S3×C3×C6, C3×C9⋊C6, C6×C9⋊C6

Smallest permutation representation of C6×C9⋊C6
►On 36 points
Generators in S36
(1 17 4 11 7 14)(2 18 5 12 8 15)(3 10 6 13 9 16)(19 31 25 28 22 34)(20 32 26 29 23 35)(21 33 27 30 24 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 28 4 34 7 31)(2 33)(3 29 9 32 6 35)(5 30)(8 36)(10 23 16 26 13 20)(11 19 14 25 17 22)(12 24)(15 21)(18 27)
 
G:=sub<Sym(36)| (1,17,4,11,7,14)(2,18,5,12,8,15)(3,10,6,13,9,16)(19,31,25,28,22,34)(20,32,26,29,23,35)(21,33,27,30,24,36), (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), (1,28,4,34,7,31)(2,33)(3,29,9,32,6,35)(5,30)(8,36)(10,23,16,26,13,20)(11,19,14,25,17,22)(12,24)(15,21)(18,27)>;
 
G:=Group( (1,17,4,11,7,14)(2,18,5,12,8,15)(3,10,6,13,9,16)(19,31,25,28,22,34)(20,32,26,29,23,35)(21,33,27,30,24,36), (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), (1,28,4,34,7,31)(2,33)(3,29,9,32,6,35)(5,30)(8,36)(10,23,16,26,13,20)(11,19,14,25,17,22)(12,24)(15,21)(18,27) );
 
G=PermutationGroup([[(1,17,4,11,7,14),(2,18,5,12,8,15),(3,10,6,13,9,16),(19,31,25,28,22,34),(20,32,26,29,23,35),(21,33,27,30,24,36)], [(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)], [(1,28,4,34,7,31),(2,33),(3,29,9,32,6,35),(5,30),(8,36),(10,23,16,26,13,20),(11,19,14,25,17,22),(12,24),(15,21),(18,27)]])
 

60 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F···3K6A6B6C6D6E6F···6K6L···6AA9A···9I18A···18I
order1222333333···3666666···66···69···918···18
size1199112223···3112223···39···96···66···6

60 irreducible representations

dim11111111122226666
type+++++++
imageC1C2C2C3C3C6C6C6C6S3D6C3×S3S3×C6C9⋊C6C2×C9⋊C6C3×C9⋊C6C6×C9⋊C6
kernelC6×C9⋊C6C3×C9⋊C6C6×3- 1+2C6×D9C2×C9⋊C6C3×D9C9⋊C6C3×C18C2×3- 1+2C32×C6C33C3×C6C32C6C3C2C1
# reps121264122611881122

Matrix representation of C6×C9⋊C6 ►in GL8(𝔽19)

80000000
08000000
00700000
00070000
00007000
00000700
00000070
00000007
,
01000000
1818000000
00070000
00007000
00100000
00000001
000001100
000000110
,
1217000000
57000000
00000700
000000110
00000001
00700000
000110000
00001000

G:=sub<GL(8,GF(19))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7],[0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0],[12,5,0,0,0,0,0,0,17,7,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C6×C9⋊C6 in GAP, Magma, Sage, TeX 

C_6\times C_9\rtimes C_6
 
% in TeX
 
G:=Group("C6xC9:C6");
 
// GroupNames label
 
G:=SmallGroup(324,140);
 
// by ID
 
G=gap.SmallGroup(324,140);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,5404,1096,208,7781]);
 
// Polycyclic
 
G:=Group<a,b,c|a^6=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
 
// generators/relations
 

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