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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, (S3×C6)⋊2D6, C3⋊1(S3×D12), Dic3⋊2S32, C32⋊9(S3×D4), C33⋊10(C2×D4), C3⋊D12⋊1S3, (C3×Dic3)⋊3D6, C6.D6⋊1S3, C32⋊9(C2×D12), C33⋊8D4⋊4C2, C3⋊1(Dic3⋊D6), (C32×C6).9C23, (C32×Dic3)⋊4C22, C2.9S33, C6.9(C2×S32), (C3×C3⋊S3)⋊4D4, (C2×C3⋊S3)⋊10D6, (S3×C3×C6)⋊6C22, (C6×C3⋊S3)⋊5C22, (C3×C3⋊D12)⋊4C2, (C3×C6.D6)⋊1C2, (C2×C32⋊4D6)⋊2C2, (C3×C6).58(C22×S3), (C2×C33⋊C2)⋊3C22, (C2×S3×C3⋊S3)⋊4C2, SmallGroup(432,602)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C3⋊S3⋊4D12
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×C3⋊D12 — C3⋊S3⋊4D12
Lower central
C33 — C32×C6 — 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, C2×C4, D4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C33, C3×Dic3, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, S3×D4, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C33⋊C2, C32×C6, C6.D6, C3⋊D12, C3⋊D12, S3×C12, C3×D12, C3×C3⋊D4, C12⋊S3, C2×S32, C22×C3⋊S3, C32×Dic3, S3×C3⋊S3, C32⋊4D6, S3×C3×C6, C6×C3⋊S3, C6×C3⋊S3, C2×C33⋊C2, S3×D12, Dic3⋊D6, C3×C6.D6, C3×C3⋊D12, C33⋊8D4, C2×S3×C3⋊S3, C2×C32⋊4D6, C3⋊S3⋊4D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, S32, C2×D12, S3×D4, C2×S32, S3×D12, 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+++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6D12S32S32S3×D4C2×S32S3×D12Dic3⋊D6S33C3⋊S3⋊4D12
kernelC3⋊S3⋊4D12C3×C6.D6C3×C3⋊D12C33⋊8D4C2×S3×C3⋊S3C2×C32⋊4D6C6.D6C3⋊D12C3×C3⋊S3C3×Dic3S3×C6C2×C3⋊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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