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## G = C6×C9⋊C6order 324 = 22·34

### Direct product of C6 and C9⋊C6

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 2A 2B 2C 3A 3B 3C 3D 3E 3F ··· 3K 6A 6B 6C 6D 6E 6F ··· 6K 6L ··· 6AA 9A ··· 9I 18A ··· 18I order 1 2 2 2 3 3 3 3 3 3 ··· 3 6 6 6 6 6 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 9 9 1 1 2 2 2 3 ··· 3 1 1 2 2 2 3 ··· 3 9 ··· 9 6 ··· 6 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 6 6 6 6 type + + + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 S3 D6 C3×S3 S3×C6 C9⋊C6 C2×C9⋊C6 C3×C9⋊C6 C6×C9⋊C6 kernel C6×C9⋊C6 C3×C9⋊C6 C6×3- 1+2 C6×D9 C2×C9⋊C6 C3×D9 C9⋊C6 C3×C18 C2×3- 1+2 C32×C6 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 2 1 2 6 4 12 2 6 1 1 8 8 1 1 2 2

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

 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 1 0 0 0 0 0 0 18 18 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 0 0 0 11 0
,
 12 17 0 0 0 0 0 0 5 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

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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