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G = C3×C9⋊C12  order 324 = 22·34

Direct product of C3 and C9⋊C12

direct product, metabelian, supersoluble, monomial

Aliases: C3×C9⋊C12, Dic9⋊C32, C33.3Dic3, 3- 1+2⋊3C12, C9⋊(C3×C12), C18.(C3×C6), (C3×C9)⋊4C12, (C3×Dic9)⋊C3, C6.6(C9⋊C6), (C3×C18).6C6, (C32×C6).8S3, C6.3(S3×C32), (C3×3- 1+2)⋊1C4, C3.3(C32×Dic3), C32.4(C3×Dic3), (C6×3- 1+2).1C2, (C2×3- 1+2).5C6, C2.(C3×C9⋊C6), (C3×C6).8(C3×S3), SmallGroup(324,94)

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 3A3B3C3D3E3F···3K4A4B6A6B6C6D6E6F···6K9A···9I12A···12P18A···18I
order12333333···344666666···69···912···1218···18
size11112223···399112223···36···69···96···6

60 irreducible representations

dim11111111122226666
type+++-+-
imageC1C2C3C3C4C6C6C12C12S3Dic3C3×S3C3×Dic3C9⋊C6C9⋊C12C3×C9⋊C6C3×C9⋊C12
kernelC3×C9⋊C12C6×3- 1+2C3×Dic9C9⋊C12C3×3- 1+2C3×C18C2×3- 1+2C3×C93- 1+2C32×C6C33C3×C6C32C6C3C2C1
# reps112622641211881122

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

100000000
010000000
002600000
000260000
000026000
000002600
000000260
000000026
,
01000000
3636000000
000100000
000010000
00100000
00000001
000002600
000000260
,
91000000
2928000000
000001000
00000010
000000026
001000000
00010000
000026000

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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