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## G = C3⋊S3×S4order 432 = 24·33

### Direct product of C3⋊S3 and S4

Aliases: C3⋊S3×S4, C628D6, (C3×S4)⋊S3, C31(S3×S4), (C3×A4)⋊4D6, C327(C2×S4), (C32×S4)⋊2C2, C324S41C2, (C32×A4)⋊2C22, (C2×C6)⋊1S32, (A4×C3⋊S3)⋊1C2, A41(C2×C3⋊S3), C222(S3×C3⋊S3), (C22×C3⋊S3)⋊6S3, SmallGroup(432,746)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3⋊S3×S4
G = < a,b,c,d,e,f,g | a3=b3=c2=d2=e2=f3=g2=1, ab=ba, cac=a-1, ad=da, ae=ea, af=fa, ag=ga, cbc=b-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 2236 in 230 conjugacy classes, 29 normal (13 characteristic)
C1, C2 [×5], C3 [×4], C3 [×5], C4 [×2], C22, C22 [×6], S3 [×18], C6 [×9], C2×C4, D4 [×4], C23 [×2], C32, C32 [×8], Dic3 [×4], C12 [×4], A4, A4 [×4], D6 [×21], C2×C6 [×4], C2×C6 [×4], C2×D4, C3×S3 [×8], C3⋊S3, C3⋊S3 [×10], C3×C6 [×2], C4×S3 [×4], D12 [×4], C3⋊D4 [×8], C3×D4 [×4], S4, S4 [×5], C2×A4, C22×S3 [×8], C33, C3⋊Dic3, C3×C12, S32 [×4], C3×A4 [×4], C3×A4 [×4], C2×C3⋊S3 [×5], C62, C62, S3×D4 [×4], C2×S4, S3×C32, C3×C3⋊S3, C33⋊C2, C4×C3⋊S3, C12⋊S3, C327D4 [×2], D4×C32, C3×S4 [×4], C3⋊S4 [×8], S3×A4 [×4], C22×C3⋊S3, C22×C3⋊S3, S3×C3⋊S3, C32×A4, D4×C3⋊S3, S3×S4 [×4], C32×S4, C324S4, A4×C3⋊S3, C3⋊S3×S4
Quotients: C1, C2 [×3], C22, S3 [×5], D6 [×5], C3⋊S3, S4, S32 [×4], C2×C3⋊S3, C2×S4, S3×C3⋊S3, S3×S4 [×4], C3⋊S3×S4

Character table of C3⋊S3×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 12A 12B 12C 12D size 1 3 6 9 27 54 2 2 2 2 8 16 16 16 16 6 54 6 6 6 6 12 12 12 12 72 12 12 12 12 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 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 -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 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 -1 -1 -1 linear of order 2 ρ5 2 2 2 0 0 0 -1 -1 2 -1 2 -1 -1 -1 2 2 0 -1 2 -1 -1 -1 -1 -1 2 0 -1 -1 2 -1 orthogonal lifted from S3 ρ6 2 2 -2 0 0 0 -1 2 -1 -1 2 -1 -1 2 -1 -2 0 2 -1 -1 -1 1 1 -2 1 0 1 -2 1 1 orthogonal lifted from D6 ρ7 2 2 -2 0 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -2 0 -1 -1 -1 2 1 -2 1 1 0 -2 1 1 1 orthogonal lifted from D6 ρ8 2 2 2 0 0 0 -1 2 -1 -1 2 -1 -1 2 -1 2 0 2 -1 -1 -1 -1 -1 2 -1 0 -1 2 -1 -1 orthogonal lifted from S3 ρ9 2 2 -2 0 0 0 -1 -1 2 -1 2 -1 -1 -1 2 -2 0 -1 2 -1 -1 1 1 1 -2 0 1 1 -2 1 orthogonal lifted from D6 ρ10 2 2 0 -2 -2 0 2 2 2 2 -1 -1 -1 -1 -1 0 0 2 2 2 2 0 0 0 0 1 0 0 0 0 orthogonal lifted from D6 ρ11 2 2 -2 0 0 0 2 -1 -1 -1 2 -1 2 -1 -1 -2 0 -1 -1 2 -1 -2 1 1 1 0 1 1 1 -2 orthogonal lifted from D6 ρ12 2 2 2 0 0 0 -1 -1 -1 2 2 2 -1 -1 -1 2 0 -1 -1 -1 2 -1 2 -1 -1 0 2 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 0 2 2 0 2 2 2 2 -1 -1 -1 -1 -1 0 0 2 2 2 2 0 0 0 0 -1 0 0 0 0 orthogonal lifted from S3 ρ14 2 2 2 0 0 0 2 -1 -1 -1 2 -1 2 -1 -1 2 0 -1 -1 2 -1 2 -1 -1 -1 0 -1 -1 -1 2 orthogonal lifted from S3 ρ15 3 -1 -1 -3 1 1 3 3 3 3 0 0 0 0 0 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 1 1 1 1 orthogonal lifted from C2×S4 ρ16 3 -1 -1 3 -1 -1 3 3 3 3 0 0 0 0 0 1 1 -1 -1 -1 -1 -1 -1 -1 -1 0 1 1 1 1 orthogonal lifted from S4 ρ17 3 -1 1 -3 1 -1 3 3 3 3 0 0 0 0 0 -1 1 -1 -1 -1 -1 1 1 1 1 0 -1 -1 -1 -1 orthogonal lifted from C2×S4 ρ18 3 -1 1 3 -1 1 3 3 3 3 0 0 0 0 0 -1 -1 -1 -1 -1 -1 1 1 1 1 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ19 4 4 0 0 0 0 4 -2 -2 -2 -2 1 -2 1 1 0 0 -2 -2 4 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ20 4 4 0 0 0 0 -2 4 -2 -2 -2 1 1 -2 1 0 0 4 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ21 4 4 0 0 0 0 -2 -2 4 -2 -2 1 1 1 -2 0 0 -2 4 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ22 4 4 0 0 0 0 -2 -2 -2 4 -2 -2 1 1 1 0 0 -2 -2 -2 4 0 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ23 6 -2 -2 0 0 0 -3 -3 -3 6 0 0 0 0 0 2 0 1 1 1 -2 1 -2 1 1 0 2 -1 -1 -1 orthogonal lifted from S3×S4 ρ24 6 -2 -2 0 0 0 6 -3 -3 -3 0 0 0 0 0 2 0 1 1 -2 1 -2 1 1 1 0 -1 -1 -1 2 orthogonal lifted from S3×S4 ρ25 6 -2 2 0 0 0 -3 -3 -3 6 0 0 0 0 0 -2 0 1 1 1 -2 -1 2 -1 -1 0 -2 1 1 1 orthogonal lifted from S3×S4 ρ26 6 -2 -2 0 0 0 -3 6 -3 -3 0 0 0 0 0 2 0 -2 1 1 1 1 1 -2 1 0 -1 2 -1 -1 orthogonal lifted from S3×S4 ρ27 6 -2 2 0 0 0 -3 -3 6 -3 0 0 0 0 0 -2 0 1 -2 1 1 -1 -1 -1 2 0 1 1 -2 1 orthogonal lifted from S3×S4 ρ28 6 -2 -2 0 0 0 -3 -3 6 -3 0 0 0 0 0 2 0 1 -2 1 1 1 1 1 -2 0 -1 -1 2 -1 orthogonal lifted from S3×S4 ρ29 6 -2 2 0 0 0 6 -3 -3 -3 0 0 0 0 0 -2 0 1 1 -2 1 2 -1 -1 -1 0 1 1 1 -2 orthogonal lifted from S3×S4 ρ30 6 -2 2 0 0 0 -3 6 -3 -3 0 0 0 0 0 -2 0 -2 1 1 1 -1 -1 2 -1 0 1 -2 1 1 orthogonal lifted from S3×S4

Smallest permutation representation of C3⋊S3×S4
On 36 points
Generators in S36
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 27 4)(2 25 5)(3 26 6)(7 17 29)(8 18 30)(9 16 28)(10 22 34)(11 23 35)(12 24 36)(13 21 33)(14 19 31)(15 20 32)
(2 3)(4 27)(5 26)(6 25)(7 28)(8 30)(9 29)(11 12)(13 14)(16 17)(19 33)(20 32)(21 31)(22 34)(23 36)(24 35)
(1 15)(2 13)(3 14)(4 32)(5 33)(6 31)(7 36)(8 34)(9 35)(10 18)(11 16)(12 17)(19 26)(20 27)(21 25)(22 30)(23 28)(24 29)
(1 18)(2 16)(3 17)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(19 24)(20 22)(21 23)(25 28)(26 29)(27 30)(31 36)(32 34)(33 35)
(7 36 31)(8 34 32)(9 35 33)(10 15 18)(11 13 16)(12 14 17)(19 29 24)(20 30 22)(21 28 23)
(10 15)(11 13)(12 14)(19 24)(20 22)(21 23)(31 36)(32 34)(33 35)

G:=sub<Sym(36)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,27,4)(2,25,5)(3,26,6)(7,17,29)(8,18,30)(9,16,28)(10,22,34)(11,23,35)(12,24,36)(13,21,33)(14,19,31)(15,20,32), (2,3)(4,27)(5,26)(6,25)(7,28)(8,30)(9,29)(11,12)(13,14)(16,17)(19,33)(20,32)(21,31)(22,34)(23,36)(24,35), (1,15)(2,13)(3,14)(4,32)(5,33)(6,31)(7,36)(8,34)(9,35)(10,18)(11,16)(12,17)(19,26)(20,27)(21,25)(22,30)(23,28)(24,29), (1,18)(2,16)(3,17)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,36)(32,34)(33,35), (7,36,31)(8,34,32)(9,35,33)(10,15,18)(11,13,16)(12,14,17)(19,29,24)(20,30,22)(21,28,23), (10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(31,36)(32,34)(33,35)>;

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)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,27,4)(2,25,5)(3,26,6)(7,17,29)(8,18,30)(9,16,28)(10,22,34)(11,23,35)(12,24,36)(13,21,33)(14,19,31)(15,20,32), (2,3)(4,27)(5,26)(6,25)(7,28)(8,30)(9,29)(11,12)(13,14)(16,17)(19,33)(20,32)(21,31)(22,34)(23,36)(24,35), (1,15)(2,13)(3,14)(4,32)(5,33)(6,31)(7,36)(8,34)(9,35)(10,18)(11,16)(12,17)(19,26)(20,27)(21,25)(22,30)(23,28)(24,29), (1,18)(2,16)(3,17)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,36)(32,34)(33,35), (7,36,31)(8,34,32)(9,35,33)(10,15,18)(11,13,16)(12,14,17)(19,29,24)(20,30,22)(21,28,23), (10,15)(11,13)(12,14)(19,24)(20,22)(21,23)(31,36)(32,34)(33,35) );

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),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,27,4),(2,25,5),(3,26,6),(7,17,29),(8,18,30),(9,16,28),(10,22,34),(11,23,35),(12,24,36),(13,21,33),(14,19,31),(15,20,32)], [(2,3),(4,27),(5,26),(6,25),(7,28),(8,30),(9,29),(11,12),(13,14),(16,17),(19,33),(20,32),(21,31),(22,34),(23,36),(24,35)], [(1,15),(2,13),(3,14),(4,32),(5,33),(6,31),(7,36),(8,34),(9,35),(10,18),(11,16),(12,17),(19,26),(20,27),(21,25),(22,30),(23,28),(24,29)], [(1,18),(2,16),(3,17),(4,8),(5,9),(6,7),(10,15),(11,13),(12,14),(19,24),(20,22),(21,23),(25,28),(26,29),(27,30),(31,36),(32,34),(33,35)], [(7,36,31),(8,34,32),(9,35,33),(10,15,18),(11,13,16),(12,14,17),(19,29,24),(20,30,22),(21,28,23)], [(10,15),(11,13),(12,14),(19,24),(20,22),(21,23),(31,36),(32,34),(33,35)])

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

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

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

C3⋊S3×S4 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times S_4
% in TeX

G:=Group("C3:S3xS4");
// GroupNames label

G:=SmallGroup(432,746);
// by ID

G=gap.SmallGroup(432,746);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,93,675,2524,4548,2287,2659,3989]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^3=b^3=c^2=d^2=e^2=f^3=g^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,c*b*c=b^-1,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,c*g=g*c,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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