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G = S3×C3⋊S4  order 432 = 24·33

Direct product of S3 and C3⋊S4

direct product, non-abelian, soluble, monomial, rational

Aliases: S3×C3⋊S4, C62⋊9D6, A4⋊1S32, (C3×S3)⋊S4, (S3×A4)⋊S3, C3⋊3(S3×S4), (C3×A4)⋊5D6, C32⋊4(C2×S4), C32⋊4S4⋊2C2, (C32×A4)⋊3C22, (C2×C6)⋊4S32, C3⋊1(C2×C3⋊S4), (C3×S3×A4)⋊2C2, (S3×C2×C6)⋊2S3, (C3×C3⋊S4)⋊1C2, C22⋊1(S3×C3⋊S3), (C22×S3)⋊(C3⋊S3), (C2×C6)⋊(C2×C3⋊S3), SmallGroup(432,747)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C32×A4 — S3×C3⋊S4
Chief series
C1 — C22 — C2×C6 — C62 — C32×A4 — C3×C3⋊S4 — S3×C3⋊S4
Lower central
C32×A4 — S3×C3⋊S4
Upper central
C1

Generators and relations for S3×C3⋊S4
 G = < a,b,c,d,e,f,g | a3=b2=c3=d2=e2=f3=g2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 2080 in 207 conjugacy classes, 28 normal (19 characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, S3, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, A4, A4, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, S4, C2×A4, C22×S3, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, S32, C3×A4, C3×A4, C3×A4, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, C2×S4, S3×C32, C3×C3⋊S3, C33⋊C2, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C32⋊7D4, C3×S4, C3⋊S4, C3⋊S4, S3×A4, C2×S32, C6×A4, S3×C2×C6, S3×C3⋊S3, C32×A4, S3×C3⋊D4, S3×S4, C2×C3⋊S4, C3×C3⋊S4, C32⋊4S4, C3×S3×A4, S3×C3⋊S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, S32, C2×C3⋊S3, C2×S4, C3⋊S4, S3×C3⋊S3, S3×S4, C2×C3⋊S4, S3×C3⋊S4

Character table of S3×C3⋊S4

 class 12A2B2C2D2E3A3B3C3D3E3F3G3H3I4A4B6A6B6C6D6E6F6G6H6I12
 size 13391854224888161616185466612182424243636
ρ1111111111111111111111111111    trivial
ρ211-1-11-11111111111-111-11-1-1-1-111    linear of order 2
ρ311-1-1-11111111111-1111-11-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111111111-1-111111111-1-1    linear of order 2
ρ52200-202-1-1222-1-1-1-202-10-1000011    orthogonal lifted from D6
ρ622-2-200-12-12-1-1-12-100-121-1111-200    orthogonal lifted from D6
ρ7222200-12-1-12-12-1-100-12-1-1-12-1-100    orthogonal lifted from S3
ρ82200202-1-1222-1-1-1202-10-10000-1-1    orthogonal lifted from S3
ρ922-2-200-12-1-1-12-1-1200-121-111-2100    orthogonal lifted from D6
ρ10222200-12-1-1-12-1-1200-12-1-1-1-12-100    orthogonal lifted from S3
ρ11222200222-1-1-1-1-1-10022222-1-1-100    orthogonal lifted from S3
ρ1222-2-200222-1-1-1-1-1-10022-22-211100    orthogonal lifted from D6
ρ1322-2-200-12-1-12-12-1-100-121-11-21100    orthogonal lifted from D6
ρ14222200-12-12-1-1-12-100-12-1-1-1-1-1200    orthogonal lifted from S3
ρ153-13-111333000000-1-1-1-13-1-10001-1    orthogonal lifted from S4
ρ163-1-31-113330000001-1-1-1-3-11000-11    orthogonal lifted from C2×S4
ρ173-13-1-1-133300000011-1-13-1-1000-11    orthogonal lifted from S4
ρ183-1-311-1333000000-11-1-1-3-110001-1    orthogonal lifted from C2×S4
ρ19440000-2-21-24-2-21100-2-201000000    orthogonal lifted from S32
ρ204400004-2-2-2-2-2111004-20-2000000    orthogonal lifted from S32
ρ21440000-2-21-2-2411-200-2-201000000    orthogonal lifted from S32
ρ22440000-2-214-2-21-2100-2-201000000    orthogonal lifted from S32
ρ236-26-200-36-3000000001-2-31100000    orthogonal lifted from C3⋊S4
ρ246-200-206-3-300000020-210100001-1    orthogonal lifted from S3×S4
ρ256-2-6200-36-3000000001-231-100000    orthogonal lifted from C2×C3⋊S4
ρ266-200206-3-3000000-20-21010000-11    orthogonal lifted from S3×S4
ρ2712-40000-6-6300000000220-1000000    orthogonal faithful

Permutation representations of S3×C3⋊S4
►On 24 points - transitive group 24T1329
Generators in S24
(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 6)(2 5)(3 4)(7 19)(8 21)(9 20)(10 17)(11 16)(12 18)(13 22)(14 24)(15 23)

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

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

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

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

(1 6)(2 4)(3 5)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 16)
 
G:=sub<Sym(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), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23), (1,2,3)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,24,23), (1,11)(2,12)(3,10)(4,17)(5,18)(6,16)(7,23)(8,24)(9,22)(13,20)(14,21)(15,19), (1,19)(2,20)(3,21)(4,8)(5,9)(6,7)(10,14)(11,15)(12,13)(16,23)(17,24)(18,22), (1,3,2)(4,5,6)(7,24,18)(8,22,16)(9,23,17)(10,20,15)(11,21,13)(12,19,14), (1,6)(2,4)(3,5)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,16)>;
 
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23), (1,2,3)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,24,23), (1,11)(2,12)(3,10)(4,17)(5,18)(6,16)(7,23)(8,24)(9,22)(13,20)(14,21)(15,19), (1,19)(2,20)(3,21)(4,8)(5,9)(6,7)(10,14)(11,15)(12,13)(16,23)(17,24)(18,22), (1,3,2)(4,5,6)(7,24,18)(8,22,16)(9,23,17)(10,20,15)(11,21,13)(12,19,14), (1,6)(2,4)(3,5)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,16) );
 
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,6),(2,5),(3,4),(7,19),(8,21),(9,20),(10,17),(11,16),(12,18),(13,22),(14,24),(15,23)], [(1,2,3),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,18,17),(19,20,21),(22,24,23)], [(1,11),(2,12),(3,10),(4,17),(5,18),(6,16),(7,23),(8,24),(9,22),(13,20),(14,21),(15,19)], [(1,19),(2,20),(3,21),(4,8),(5,9),(6,7),(10,14),(11,15),(12,13),(16,23),(17,24),(18,22)], [(1,3,2),(4,5,6),(7,24,18),(8,22,16),(9,23,17),(10,20,15),(11,21,13),(12,19,14)], [(1,6),(2,4),(3,5),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,16)]])
 
G:=TransitiveGroup(24,1329);
 

Matrix representation of S3×C3⋊S4 ►in GL7(ℤ)

1000000
0100000
00-11000
00-10000
0000100
0000010
0000001
,
-1000000
0-100000
000-1000
00-10000
0000100
0000010
0000001
,
-1-100000
1000000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
0000001
00001-11
0000100
,
1000000
0100000
0010000
0001000
0000-11-1
000000-1
00000-10
,
0100000
-1-100000
0010000
0001000
00000-10
000000-1
0000100
,
-1000000
1100000
00-10000
000-1000
0000-100
0000001
0000010

G:=sub<GL(7,Integers())| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1,0,0,0,0,0,1,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,-1,0,0,0,0,-1,-1,0],[0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0],[-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;
 

S3×C3⋊S4 in GAP, Magma, Sage, TeX 

S_3\times C_3\rtimes S_4
 
% in TeX
 
G:=Group("S3xC3:S4");
 
// GroupNames label
 
G:=SmallGroup(432,747);
 
// by ID
 
G=gap.SmallGroup(432,747);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,170,346,2524,9077,2287,5298,3989]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^3=d^2=e^2=f^3=g^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,g*c*g=c^-1,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;
 
// generators/relations
 

Export

Character table of S3×C3⋊S4 in TeX

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