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## G = S3×C3⋊S4order 432 = 24·33

### Direct product of S3 and C3⋊S4

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 [×5], C3 [×2], C3 [×7], C4 [×2], C22, C22 [×6], S3, S3 [×14], C6 [×9], C2×C4, D4 [×4], C23 [×2], C32, C32 [×8], Dic3 [×4], C12, A4 [×3], A4 [×3], D6 [×13], C2×C6 [×2], C2×C6 [×4], C2×D4, C3×S3, C3×S3 [×8], C3⋊S3 [×10], C3×C6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4 [×7], C3×D4, S4 [×9], C2×A4 [×3], C22×S3, C22×S3 [×2], C22×C6, C33, C3×Dic3, C3⋊Dic3, S32 [×5], C3×A4, C3×A4 [×3], C3×A4 [×4], S3×C6 [×3], C2×C3⋊S3 [×2], C62, S3×D4, C2×C3⋊D4, C2×S4 [×3], S3×C32, C3×C3⋊S3, C33⋊C2, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C327D4, C3×S4 [×3], C3⋊S4, C3⋊S4 [×8], S3×A4 [×3], C2×S32, C6×A4, S3×C2×C6, S3×C3⋊S3, C32×A4, S3×C3⋊D4, S3×S4 [×3], C2×C3⋊S4, C3×C3⋊S4, C324S4, C3×S3×A4, S3×C3⋊S4
Quotients: C1, C2 [×3], C22, S3 [×5], D6 [×5], C3⋊S3, S4, S32 [×4], 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 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 3F 3G 3H 3I 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 12 size 1 3 3 9 18 54 2 2 4 8 8 8 16 16 16 18 54 6 6 6 12 18 24 24 24 36 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ5 2 2 0 0 -2 0 2 -1 -1 2 2 2 -1 -1 -1 -2 0 2 -1 0 -1 0 0 0 0 1 1 orthogonal lifted from D6 ρ6 2 2 -2 -2 0 0 -1 2 -1 2 -1 -1 -1 2 -1 0 0 -1 2 1 -1 1 1 1 -2 0 0 orthogonal lifted from D6 ρ7 2 2 2 2 0 0 -1 2 -1 -1 2 -1 2 -1 -1 0 0 -1 2 -1 -1 -1 2 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 2 0 0 2 0 2 -1 -1 2 2 2 -1 -1 -1 2 0 2 -1 0 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ9 2 2 -2 -2 0 0 -1 2 -1 -1 -1 2 -1 -1 2 0 0 -1 2 1 -1 1 1 -2 1 0 0 orthogonal lifted from D6 ρ10 2 2 2 2 0 0 -1 2 -1 -1 -1 2 -1 -1 2 0 0 -1 2 -1 -1 -1 -1 2 -1 0 0 orthogonal lifted from S3 ρ11 2 2 2 2 0 0 2 2 2 -1 -1 -1 -1 -1 -1 0 0 2 2 2 2 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ12 2 2 -2 -2 0 0 2 2 2 -1 -1 -1 -1 -1 -1 0 0 2 2 -2 2 -2 1 1 1 0 0 orthogonal lifted from D6 ρ13 2 2 -2 -2 0 0 -1 2 -1 -1 2 -1 2 -1 -1 0 0 -1 2 1 -1 1 -2 1 1 0 0 orthogonal lifted from D6 ρ14 2 2 2 2 0 0 -1 2 -1 2 -1 -1 -1 2 -1 0 0 -1 2 -1 -1 -1 -1 -1 2 0 0 orthogonal lifted from S3 ρ15 3 -1 3 -1 1 1 3 3 3 0 0 0 0 0 0 -1 -1 -1 -1 3 -1 -1 0 0 0 1 -1 orthogonal lifted from S4 ρ16 3 -1 -3 1 -1 1 3 3 3 0 0 0 0 0 0 1 -1 -1 -1 -3 -1 1 0 0 0 -1 1 orthogonal lifted from C2×S4 ρ17 3 -1 3 -1 -1 -1 3 3 3 0 0 0 0 0 0 1 1 -1 -1 3 -1 -1 0 0 0 -1 1 orthogonal lifted from S4 ρ18 3 -1 -3 1 1 -1 3 3 3 0 0 0 0 0 0 -1 1 -1 -1 -3 -1 1 0 0 0 1 -1 orthogonal lifted from C2×S4 ρ19 4 4 0 0 0 0 -2 -2 1 -2 4 -2 -2 1 1 0 0 -2 -2 0 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ20 4 4 0 0 0 0 4 -2 -2 -2 -2 -2 1 1 1 0 0 4 -2 0 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ21 4 4 0 0 0 0 -2 -2 1 -2 -2 4 1 1 -2 0 0 -2 -2 0 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ22 4 4 0 0 0 0 -2 -2 1 4 -2 -2 1 -2 1 0 0 -2 -2 0 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ23 6 -2 6 -2 0 0 -3 6 -3 0 0 0 0 0 0 0 0 1 -2 -3 1 1 0 0 0 0 0 orthogonal lifted from C3⋊S4 ρ24 6 -2 0 0 -2 0 6 -3 -3 0 0 0 0 0 0 2 0 -2 1 0 1 0 0 0 0 1 -1 orthogonal lifted from S3×S4 ρ25 6 -2 -6 2 0 0 -3 6 -3 0 0 0 0 0 0 0 0 1 -2 3 1 -1 0 0 0 0 0 orthogonal lifted from C2×C3⋊S4 ρ26 6 -2 0 0 2 0 6 -3 -3 0 0 0 0 0 0 -2 0 -2 1 0 1 0 0 0 0 -1 1 orthogonal lifted from S3×S4 ρ27 12 -4 0 0 0 0 -6 -6 3 0 0 0 0 0 0 0 0 2 2 0 -1 0 0 0 0 0 0 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)

 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 0 1 0 0 0 0 1 -1 1 0 0 0 0 1 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 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -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 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0
,
 -1 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 0 0 0 0 0 1 0

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

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