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## G = S3×C22⋊A4order 288 = 25·32

### Direct product of S3 and C22⋊A4

Aliases: S3×C22⋊A4, (C23×C6)⋊6C6, C222(S3×A4), C249(C3×S3), (S3×C24)⋊2C3, (C22×S3)⋊2A4, (C2×C6)⋊(C2×A4), C3⋊(C2×C22⋊A4), (C3×C22⋊A4)⋊5C2, SmallGroup(288,1038)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C6 — S3×C22⋊A4
 Chief series C1 — C3 — C2×C6 — C23×C6 — C3×C22⋊A4 — S3×C22⋊A4
 Lower central C23×C6 — S3×C22⋊A4
 Upper central C1

Generators and relations for S3×C22⋊A4
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=g3=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, gdg-1=c, geg-1=ef=fe, gfg-1=e >

Subgroups: 1658 in 303 conjugacy classes, 24 normal (9 characteristic)
C1, C2, C3, C3, C22, C22, S3, S3, C6, C23, C32, A4, D6, C2×C6, C2×C6, C24, C24, C3×S3, C2×A4, C22×S3, C22×S3, C22×C6, C25, C3×A4, C22⋊A4, C22⋊A4, S3×C23, C23×C6, S3×A4, C2×C22⋊A4, S3×C24, C3×C22⋊A4, S3×C22⋊A4
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, C22⋊A4, S3×A4, C2×C22⋊A4, S3×C22⋊A4

Character table of S3×C22⋊A4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G size 1 3 3 3 3 3 3 9 9 9 9 9 2 16 16 32 32 6 6 6 6 6 48 48 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 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 linear of order 2 ρ3 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 ζ65 ζ6 linear of order 6 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 ζ32 ζ3 linear of order 3 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 ζ3 ζ32 linear of order 3 ρ6 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 ζ6 ζ65 linear of order 6 ρ7 2 2 2 0 2 2 2 0 0 0 0 0 -1 2 2 -1 -1 -1 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 2 2 0 2 2 2 0 0 0 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 -1 -1 -1 -1 -1 0 0 complex lifted from C3×S3 ρ9 2 2 2 0 2 2 2 0 0 0 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 -1 -1 -1 -1 -1 0 0 complex lifted from C3×S3 ρ10 3 -1 -1 3 -1 -1 3 -1 3 -1 -1 -1 3 0 0 0 0 -1 -1 -1 3 -1 0 0 orthogonal lifted from A4 ρ11 3 -1 3 3 -1 -1 -1 -1 -1 3 -1 -1 3 0 0 0 0 -1 -1 -1 -1 3 0 0 orthogonal lifted from A4 ρ12 3 3 -1 -3 -1 -1 -1 1 1 1 -3 1 3 0 0 0 0 3 -1 -1 -1 -1 0 0 orthogonal lifted from C2×A4 ρ13 3 -1 -1 -3 -1 3 -1 -3 1 1 1 1 3 0 0 0 0 -1 3 -1 -1 -1 0 0 orthogonal lifted from C2×A4 ρ14 3 -1 -1 3 3 -1 -1 -1 -1 -1 -1 3 3 0 0 0 0 -1 -1 3 -1 -1 0 0 orthogonal lifted from A4 ρ15 3 -1 -1 -3 3 -1 -1 1 1 1 1 -3 3 0 0 0 0 -1 -1 3 -1 -1 0 0 orthogonal lifted from C2×A4 ρ16 3 -1 -1 -3 -1 -1 3 1 -3 1 1 1 3 0 0 0 0 -1 -1 -1 3 -1 0 0 orthogonal lifted from C2×A4 ρ17 3 -1 3 -3 -1 -1 -1 1 1 -3 1 1 3 0 0 0 0 -1 -1 -1 -1 3 0 0 orthogonal lifted from C2×A4 ρ18 3 3 -1 3 -1 -1 -1 -1 -1 -1 3 -1 3 0 0 0 0 3 -1 -1 -1 -1 0 0 orthogonal lifted from A4 ρ19 3 -1 -1 3 -1 3 -1 3 -1 -1 -1 -1 3 0 0 0 0 -1 3 -1 -1 -1 0 0 orthogonal lifted from A4 ρ20 6 -2 -2 0 -2 6 -2 0 0 0 0 0 -3 0 0 0 0 1 -3 1 1 1 0 0 orthogonal lifted from S3×A4 ρ21 6 6 -2 0 -2 -2 -2 0 0 0 0 0 -3 0 0 0 0 -3 1 1 1 1 0 0 orthogonal lifted from S3×A4 ρ22 6 -2 -2 0 6 -2 -2 0 0 0 0 0 -3 0 0 0 0 1 1 -3 1 1 0 0 orthogonal lifted from S3×A4 ρ23 6 -2 -2 0 -2 -2 6 0 0 0 0 0 -3 0 0 0 0 1 1 1 -3 1 0 0 orthogonal lifted from S3×A4 ρ24 6 -2 6 0 -2 -2 -2 0 0 0 0 0 -3 0 0 0 0 1 1 1 1 -3 0 0 orthogonal lifted from S3×A4

Smallest permutation representation of S3×C22⋊A4
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 7)(2 9)(3 8)(4 10)(5 12)(6 11)(13 19)(14 21)(15 20)(16 22)(17 24)(18 23)(25 31)(26 33)(27 32)(28 34)(29 36)(30 35)
(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)

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

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

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

Matrix representation of S3×C22⋊A4 in GL8(ℤ)

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

G:=sub<GL(8,Integers())| [0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

S3×C22⋊A4 in GAP, Magma, Sage, TeX

S_3\times C_2^2\rtimes A_4
% in TeX

G:=Group("S3xC2^2:A4");
// GroupNames label

G:=SmallGroup(288,1038);
// by ID

G=gap.SmallGroup(288,1038);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-3,198,94,1271,516,9414]);
// Polycyclic

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

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