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G = C3:S3:4D12  order 432 = 24·33

The semidirect product of C3:S3 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D6:6S32, C3:S3:4D12, (S3xC6):2D6, C3:1(S3xD12), Dic3:2S32, C32:9(S3xD4), C33:10(C2xD4), C3:D12:1S3, (C3xDic3):3D6, C6.D6:1S3, C32:9(C2xD12), C33:8D4:4C2, C3:1(Dic3:D6), (C32xC6).9C23, (C32xDic3):4C22, C2.9S33, C6.9(C2xS32), (C3xC3:S3):4D4, (C2xC3:S3):10D6, (S3xC3xC6):6C22, (C6xC3:S3):5C22, (C3xC3:D12):4C2, (C3xC6.D6):1C2, (C2xC32:4D6):2C2, (C3xC6).58(C22xS3), (C2xC33:C2):3C22, (C2xS3xC3:S3):4C2, SmallGroup(432,602)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — C3:S3:4D12
Chief series
C1 — C3 — C32 — C33 — C32xC6 — S3xC3xC6 — C3xC3:D12 — C3:S3:4D12
Lower central
C33 — C32xC6 — C3:S3:4D12
Upper central
C1 — C2

Generators and relations for C3:S3:4D12
 G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2220 in 290 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2xC4, D4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2xC6, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C22xS3, C33, C3xDic3, C3xDic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C2xC3:S3, C62, C2xD12, S3xD4, S3xC32, C3xC3:S3, C3xC3:S3, C33:C2, C32xC6, C6.D6, C3:D12, C3:D12, S3xC12, C3xD12, C3xC3:D4, C12:S3, C2xS32, C22xC3:S3, C32xDic3, S3xC3:S3, C32:4D6, S3xC3xC6, C6xC3:S3, C6xC3:S3, C2xC33:C2, S3xD12, Dic3:D6, C3xC6.D6, C3xC3:D12, C33:8D4, C2xS3xC3:S3, C2xC32:4D6, C3:S3:4D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, S32, C2xD12, S3xD4, C2xS32, S3xD12, Dic3:D6, S33, C3:S3:4D12

Permutation representations of C3:S3:4D12
►On 24 points - transitive group 24T1297
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 21 17)(14 18 22)(15 23 19)(16 20 24)

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

(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)
 
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)(13,21,17)(14,18,22)(15,23,19)(16,20,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)>;
 
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)(13,21,17)(14,18,22)(15,23,19)(16,20,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19) );
 
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),(13,21,17),(14,18,22),(15,23,19),(16,20,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19)]])
 
G:=TransitiveGroup(24,1297);
 

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F3G4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O12A12B12C12D12E···12J
order122222223333333446666666666666661212121212···12
size1169918185422244486622244481212121218183636666612···12

42 irreducible representations

dim111111222222244444488
type+++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6D12S32S32S3xD4C2xS32S3xD12Dic3:D6S33C3:S3:4D12
kernelC3:S3:4D12C3xC6.D6C3xC3:D12C33:8D4C2xS3xC3:S3C2xC32:4D6C6.D6C3:D12C3xC3:S3C3xDic3S3xC6C2xC3:S3C3:S3Dic3D6C32C6C3C3C2C1
# reps112211122423421234211

Matrix representation of C3:S3:4D12 ►in GL8(Z)

-1-1000000
10000000
00-1-10000
00100000
00000100
0000-1-100
00000001
000000-1-1
,
01000000
-1-1000000
00-1-10000
00100000
00000100
0000-1-100
000000-1-1
00000010
,
00000010
00000001
0000-1000
00000-100
00-100000
000-10000
10000000
01000000
,
00-1-10000
00100000
11000000
-10000000
000000-1-1
00000010
00001100
0000-1000
,
00001100
00000-100
000000-1-1
00000001
11000000
0-1000000
00-1-10000
00010000

G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0] >;
 

C3:S3:4D12 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3\rtimes_4D_{12}
 
% in TeX
 
G:=Group("C3:S3:4D12");
 
// GroupNames label
 
G:=SmallGroup(432,602);
 
// by ID
 
G=gap.SmallGroup(432,602);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,254,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=e^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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