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G = C3×C32⋊C12  order 324 = 22·34

Direct product of C3 and C32⋊C12

direct product, metabelian, supersoluble, monomial

Aliases: C3×C32⋊C12, He3⋊5C12, C33⋊2C12, C33⋊3Dic3, C32⋊(C3×C12), (C3×He3)⋊3C4, C3⋊Dic3⋊C32, (C6×He3).1C2, C6.2(S3×C32), (C32×C6).7S3, (C32×C6).4C6, (C2×He3).10C6, C32⋊1(C3×Dic3), C6.18(C32⋊C6), C3.2(C32×Dic3), (C3×C6).(C3×C6), (C3×C3⋊Dic3)⋊C3, C2.(C3×C32⋊C6), (C3×C6).7(C3×S3), SmallGroup(324,92)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C32⋊C12
Chief series
C1 — C3 — C32 — C3×C6 — C32×C6 — C6×He3 — C3×C32⋊C12
Lower central
C32 — C3×C32⋊C12
Upper central
C1 — C6

Generators and relations for C3×C32⋊C12
 G = < a,b,c,d | a3=b3=c3=d12=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=c-1 >

Subgroups: 304 in 96 conjugacy classes, 34 normal (19 characteristic)
C1, C2, C3, C3, C4, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, He3, He3, C33, C33, C3×Dic3, C3⋊Dic3, C3×C12, C2×He3, C2×He3, C32×C6, C32×C6, C3×He3, C32⋊C12, C32×Dic3, C3×C3⋊Dic3, C6×He3, C3×C32⋊C12
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, C3×Dic3, C3×C12, C32⋊C6, S3×C32, C32⋊C12, C32×Dic3, C3×C32⋊C6, C3×C32⋊C12

Smallest permutation representation of C3×C32⋊C12
►On 36 points
Generators in S36
(1 8 9)(2 5 10)(3 6 11)(4 7 12)(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 24 28)(2 25 21)(3 18 34)(4 31 15)(5 29 17)(6 14 26)(7 35 23)(8 20 32)(9 16 36)(10 33 13)(11 22 30)(12 27 19)

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

(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)| (1,8,9)(2,5,10)(3,6,11)(4,7,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36), (1,24,28)(2,25,21)(3,18,34)(4,31,15)(5,29,17)(6,14,26)(7,35,23)(8,20,32)(9,16,36)(10,33,13)(11,22,30)(12,27,19), (1,9,8)(2,5,10)(3,11,6)(4,7,12)(13,21,17)(14,18,22)(15,23,19)(16,20,24)(25,29,33)(26,34,30)(27,31,35)(28,36,32), (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( (1,8,9)(2,5,10)(3,6,11)(4,7,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36), (1,24,28)(2,25,21)(3,18,34)(4,31,15)(5,29,17)(6,14,26)(7,35,23)(8,20,32)(9,16,36)(10,33,13)(11,22,30)(12,27,19), (1,9,8)(2,5,10)(3,11,6)(4,7,12)(13,21,17)(14,18,22)(15,23,19)(16,20,24)(25,29,33)(26,34,30)(27,31,35)(28,36,32), (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([[(1,8,9),(2,5,10),(3,6,11),(4,7,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36)], [(1,24,28),(2,25,21),(3,18,34),(4,31,15),(5,29,17),(6,14,26),(7,35,23),(8,20,32),(9,16,36),(10,33,13),(11,22,30),(12,27,19)], [(1,9,8),(2,5,10),(3,11,6),(4,7,12),(13,21,17),(14,18,22),(15,23,19),(16,20,24),(25,29,33),(26,34,30),(27,31,35),(28,36,32)], [(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···3K3L···3T4A4B6A6B6C6D6E6F···6K6L···6T12A···12P
order12333333···33···344666666···66···612···12
size11112223···36···699112223···36···69···9

60 irreducible representations

dim11111111122226666
type+++-+-
imageC1C2C3C3C4C6C6C12C12S3Dic3C3×S3C3×Dic3C32⋊C6C32⋊C12C3×C32⋊C6C3×C32⋊C12
kernelC3×C32⋊C12C6×He3C32⋊C12C3×C3⋊Dic3C3×He3C2×He3C32×C6He3C33C32×C6C33C3×C6C32C6C3C2C1
# reps116226212411881122

Matrix representation of C3×C32⋊C12 ►in GL8(𝔽13)

90000000
09000000
00300000
00030000
00003000
00000300
00000030
00000003
,
121000000
120000000
00090303
00009030
00900101010
00000300
0000061010
00000030
,
10000000
01000000
00300033
000300010
000030100
00000900
00000090
00000009
,
08000000
80000000
000193312
00004090
000120004
003190312
00020001
00005090

G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,10,3,6,0,0,0,0,3,10,0,10,3,0,0,3,0,10,0,10,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,3,0,10,0,9,0,0,0,3,10,0,0,0,9],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,1,0,12,1,2,0,0,0,9,4,0,9,0,5,0,0,3,0,0,0,0,0,0,0,3,9,0,3,0,9,0,0,12,0,4,12,1,0] >;
 

C3×C32⋊C12 in GAP, Magma, Sage, TeX 

C_3\times C_3^2\rtimes C_{12}
 
% in TeX
 
G:=Group("C3xC3^2:C12");
 
// GroupNames label
 
G:=SmallGroup(324,92);
 
// by ID
 
G=gap.SmallGroup(324,92);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,2164,2170,7781]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=c^-1>;
 
// generators/relations
 

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