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## G = C3×C9⋊C12order 324 = 22·34

### Direct product of C3 and C9⋊C12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C3×C9⋊C12
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×3- 1+2 — C3×C9⋊C12
 Lower central C9 — C3×C9⋊C12
 Upper central C1 — C6

Generators and relations for C3×C9⋊C12
G = < a,b,c | a3=b9=c12=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 178 in 74 conjugacy classes, 34 normal (19 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C9, C32, C32, C32, Dic3, C12, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, Dic9, C3×Dic3, C3×C12, C3×C18, C3×C18, C2×3- 1+2, C2×3- 1+2, C32×C6, C3×3- 1+2, C3×Dic9, C9⋊C12, C32×Dic3, C6×3- 1+2, C3×C9⋊C12
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, C3×Dic3, C3×C12, C9⋊C6, S3×C32, C9⋊C12, C32×Dic3, C3×C9⋊C6, C3×C9⋊C12

Smallest permutation representation of C3×C9⋊C12
On 36 points
Generators in S36
(1 6 10)(2 7 11)(3 8 12)(4 5 9)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)
(1 31 18 10 27 22 6 35 14)(2 23 32 7 19 36 11 15 28)(3 25 24 12 33 16 8 29 20)(4 17 26 5 13 30 9 21 34)
(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)

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

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

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

60 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 4A 4B 6A 6B 6C 6D 6E 6F ··· 6K 9A ··· 9I 12A ··· 12P 18A ··· 18I order 1 2 3 3 3 3 3 3 ··· 3 4 4 6 6 6 6 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 1 1 2 2 2 3 ··· 3 9 9 1 1 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9 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 C3 C3 C4 C6 C6 C12 C12 S3 Dic3 C3×S3 C3×Dic3 C9⋊C6 C9⋊C12 C3×C9⋊C6 C3×C9⋊C12 kernel C3×C9⋊C12 C6×3- 1+2 C3×Dic9 C9⋊C12 C3×3- 1+2 C3×C18 C2×3- 1+2 C3×C9 3- 1+2 C32×C6 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 6 2 2 6 4 12 1 1 8 8 1 1 2 2

Matrix representation of C3×C9⋊C12 in GL8(𝔽37)

 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 26
,
 0 1 0 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 26 0 0 0 0 0 0 0 0 26 0
,
 9 1 0 0 0 0 0 0 29 28 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 26 0 0 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 26 0 0 0

G:=sub<GL(8,GF(37))| [10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,26],[0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,26,0,0,0,0,0,1,0,0],[9,29,0,0,0,0,0,0,1,28,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,26,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,26,0,0,0] >;

C3×C9⋊C12 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes C_{12}
% in TeX

G:=Group("C3xC9:C12");
// GroupNames label

G:=SmallGroup(324,94);
// by ID

G=gap.SmallGroup(324,94);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,5404,2170,208,7781]);
// Polycyclic

G:=Group<a,b,c|a^3=b^9=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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