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G = C3:S3.2D12  order 432 = 24·33

1st non-split extension by C3:S3 of D12 acting via D12/C6=C22

non-abelian, soluble, monomial

Aliases: C6.12S3wrC2, C3:S3.2D12, C6.D6:4S3, (C32xC6).6D4, C32:3(D6:C4), C32:4D6:1C4, C33:3(C22:C4), C2.1(C33:D4), C3:1(S32:C4), C3:S3.2(C4xS3), (C3xC3:S3).9D4, (C2xC3:S3).10D6, (C2xC33:C4):2C2, (C3xC6.D6):8C2, (C6xC3:S3).6C22, (C3xC6).12(C3:D4), (C2xC32:4D6).1C2, (C3xC3:S3).8(C2xC4), SmallGroup(432,579)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3xC3:S3 — C3:S3.2D12
Chief series
C1 — C3 — C33 — C3xC3:S3 — C6xC3:S3 — C2xC32:4D6 — C3:S3.2D12
Lower central
C33 — C3xC3:S3 — C3:S3.2D12
Upper central
C1 — C2

Generators and relations for C3:S3.2D12
 G = < a,b,c,d,e | a3=b3=c2=d12=1, e2=c, ab=ba, cac=ebe-1=a-1, ad=da, eae-1=b, cbc=dbd-1=b-1, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 1012 in 132 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2xC4, C23, C32, C32, Dic3, C12, D6, C2xC6, C22:C4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C22xS3, C33, C3xDic3, C3xC12, C32:C4, S32, S3xC6, C2xC3:S3, C2xC3:S3, D6:C4, C3xC3:S3, C3xC3:S3, C32xC6, C6.D6, S3xC12, C2xC32:C4, C2xS32, C32xDic3, C33:C4, C32:4D6, C32:4D6, C6xC3:S3, C6xC3:S3, S32:C4, C3xC6.D6, C2xC33:C4, C2xC32:4D6, C3:S3.2D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, C4xS3, D12, C3:D4, D6:C4, S3wrC2, S32:C4, C33:D4, C3:S3.2D12

Permutation representations of C3:S3.2D12
►On 24 points - transitive group 24T1313
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)

(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 17 21)(14 22 18)(15 19 23)(16 24 20)

(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)

(1 13 20 6)(2 5 21 24)(3 23 22 4)(7 19 14 12)(8 11 15 18)(9 17 16 10)
 
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,13,20,6)(2,5,21,24)(3,23,22,4)(7,19,14,12)(8,11,15,18)(9,17,16,10)>;
 
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,13,20,6)(2,5,21,24)(3,23,22,4)(7,19,14,12)(8,11,15,18)(9,17,16,10) );
 
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,5,9),(2,10,6),(3,7,11),(4,12,8),(13,17,21),(14,22,18),(15,19,23),(16,24,20)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,13,20,6),(2,5,21,24),(3,23,22,4),(7,19,14,12),(8,11,15,18),(9,17,16,10)]])
 
G:=TransitiveGroup(24,1313);
 

36 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E3F4A4B4C4D6A6B6C6D6E6F6G6H6I6J12A12B12C12D12E···12J
order122222333333444466666666661212121212···12
size1199181824444866545424444818183636666612···12

36 irreducible representations

dim1111122222224444488
type+++++++++++++
imageC1C2C2C2C4S3D4D4D6C4xS3D12C3:D4S3wrC2S32:C4S32:C4C33:D4C3:S3.2D12C33:D4C3:S3.2D12
kernelC3:S3.2D12C3xC6.D6C2xC33:C4C2xC32:4D6C32:4D6C6.D6C3xC3:S3C32xC6C2xC3:S3C3:S3C3:S3C3xC6C6C3C3C2C1C2C1
# reps1111411112224224411

Matrix representation of C3:S3.2D12 ►in GL6(F13)

100000
010000
00120120
000001
001000
00012012
,
100000
010000
000010
000001
00120120
00012012
,
100000
010000
001000
000100
00120120
00012012
,
10100000
370000
0001200
0012000
000101
001010
,
1060000
330000
000100
001000
00012012
000010

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,12,0,0,0,0,1,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,12,0,0,0,0,1,0,12,0,0,0,0,12,0,0,0,0,0,0,12],[10,3,0,0,0,0,10,7,0,0,0,0,0,0,0,12,0,1,0,0,12,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,3,0,0,0,0,6,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,12,0,0,0,0,0,0,1,0,0,0,0,12,0] >;
 

C3:S3.2D12 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3._2D_{12}
 
% in TeX
 
G:=Group("C3:S3.2D12");
 
// GroupNames label
 
G:=SmallGroup(432,579);
 
// by ID
 
G=gap.SmallGroup(432,579);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,1684,571,298,677,1027,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=1,e^2=c,a*b=b*a,c*a*c=e*b*e^-1=a^-1,a*d=d*a,e*a*e^-1=b,c*b*c=d*b*d^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
 
// generators/relations
 

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