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G = S3×C3⋊D12  order 432 = 24·33

Direct product of S3 and C3⋊D12

direct product, metabelian, supersoluble, monomial

Aliases: S3×C3⋊D12, D6⋊2S32, (S3×C6)⋊8D6, C3⋊5(S3×D12), Dic3⋊1S32, C33⋊6(C2×D4), (C3×S3)⋊2D12, C3⋊Dic3⋊8D6, (S3×Dic3)⋊1S3, (C3×Dic3)⋊6D6, (S3×C32)⋊3D4, C32⋊14(S3×D4), C32⋊8(C2×D12), C33⋊8D4⋊3C2, C33⋊7D4⋊1C2, C33⋊9D4⋊2C2, (C32×C6).5C23, (C32×Dic3)⋊1C22, C2.5S33, (C2×S32)⋊2S3, (S32×C6)⋊2C2, C6.5(C2×S32), C3⋊1(S3×C3⋊D4), (C2×C3⋊S3)⋊12D6, (S3×C3×C6)⋊2C22, (C3×S3×Dic3)⋊3C2, C3⋊1(C2×C3⋊D12), (C6×C3⋊S3)⋊2C22, C32⋊7(C2×C3⋊D4), (C3×C3⋊D12)⋊1C2, (C3×S3)⋊1(C3⋊D4), (C3×C6).54(C22×S3), (C3×C3⋊Dic3)⋊1C22, (C2×C33⋊C2)⋊1C22, (C2×S3×C3⋊S3)⋊1C2, SmallGroup(432,598)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — S3×C3⋊D12
Chief series
C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S32×C6 — S3×C3⋊D12
Lower central
C33 — C32×C6 — S3×C3⋊D12
Upper central
C1 — C2

Generators and relations for S3×C3⋊D12
 G = < a,b,c,d,e | a3=b2=c3=d12=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 2052 in 290 conjugacy classes, 54 normal (46 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, S3, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, S3×D4, C2×C3⋊D4, S3×C32, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, S3×Dic3, D6⋊S3, C3⋊D12, C3⋊D12, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C12⋊S3, C32⋊7D4, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C3×S32, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, S3×D12, C2×C3⋊D12, S3×C3⋊D4, C3×S3×Dic3, C3×C3⋊D12, C33⋊7D4, C33⋊8D4, C33⋊9D4, S32×C6, C2×S3×C3⋊S3, S3×C3⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, S32, C2×D12, S3×D4, C2×C3⋊D4, C3⋊D12, C2×S32, S3×D12, C2×C3⋊D12, S3×C3⋊D4, S33, S3×C3⋊D12

Permutation representations of S3×C3⋊D12
►On 24 points - transitive group 24T1295
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)

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

(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 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)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15), (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,21,17)(14,18,22)(15,23,19)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(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( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15), (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,21,17)(14,18,22)(15,23,19)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(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([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15)], [(1,5,9),(2,10,6),(3,7,11),(4,12,8),(13,21,17),(14,18,22),(15,23,19),(16,20,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(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(24,1295);
 

45 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F3G4A4B6A6B6C6D6E6F6G···6L6M6N···6R6S6T6U12A12B12C12D12E12F12G
order122222223333333446666666···666···666612121212121212
size1133618185422244486182224446···6812···12181836661212121818

45 irreducible representations

dim111111112222222222444444488
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3S3D4D6D6D6D6D12C3⋊D4S32S32S3×D4C3⋊D12C2×S32S3×D12S3×C3⋊D4S33S3×C3⋊D12
kernelS3×C3⋊D12C3×S3×Dic3C3×C3⋊D12C33⋊7D4C33⋊8D4C33⋊9D4S32×C6C2×S3×C3⋊S3S3×Dic3C3⋊D12C2×S32S3×C32C3×Dic3C3⋊Dic3S3×C6C2×C3⋊S3C3×S3C3×S3Dic3D6C32S3C6C3C3C2C1
# reps111111111112214244121232211

Matrix representation of S3×C3⋊D12 ►in GL8(ℤ)

-1-1000000
10000000
00-1-10000
00100000
0000-1-100
00001000
000000-1-1
00000010
,
00000010
000000-1-1
00001000
0000-1-100
00100000
00-1-10000
10000000
-1-1000000
,
-1-1000000
10000000
00010000
00-1-10000
0000-1-100
00001000
00000001
000000-1-1
,
00010000
00-1-10000
0-1000000
11000000
00000011
000000-10
0000-1-100
00001000
,
00000011
000000-10
0000-1-100
00001000
00010000
00-1-10000
0-1000000
11000000

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,-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],[0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[-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,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-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],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,-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] >;
 

S3×C3⋊D12 in GAP, Magma, Sage, TeX 

S_3\times C_3\rtimes D_{12}
 
% in TeX
 
G:=Group("S3xC3:D12");
 
// GroupNames label
 
G:=SmallGroup(432,598);
 
// by ID
 
G=gap.SmallGroup(432,598);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^12=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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