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## G = (C22×S3)⋊A4order 288 = 25·32

### The semidirect product of C22×S3 and A4 acting faithfully

Aliases: (C22×S3)⋊A4, C244S3⋊C3, C22⋊A43S3, (C23×C6)⋊3C6, C246(C3×S3), C3⋊(C24⋊C6), C22.4(S3×A4), (C2×C6).4(C2×A4), (C3×C22⋊A4)⋊3C2, SmallGroup(288,411)

Series: Derived Chief Lower central Upper central

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

Generators and relations for (C22×S3)⋊A4
G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e2=f2=g3=1, gbg-1=ab=ba, ac=ca, fdf=gdg-1=ad=da, ae=ea, af=fa, gag-1=b, bc=cb, ede=bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, cg=gc, geg-1=ef=fe, gfg-1=e >

Subgroups: 522 in 77 conjugacy classes, 11 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, A4, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C2×Dic3, C3⋊D4, C2×A4, C22×S3, C22×C6, C22≀C2, C3×A4, C6.D4, C2×C3⋊D4, C22⋊A4, C22⋊A4, C23×C6, S3×A4, C24⋊C6, C244S3, C3×C22⋊A4, (C22×S3)⋊A4
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, S3×A4, C24⋊C6, (C22×S3)⋊A4

Character table of (C22×S3)⋊A4

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F 6G size 1 3 6 6 12 2 16 16 32 32 36 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 trivial ρ2 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 ζ32 ζ3 ζ3 ζ32 -1 1 1 1 1 1 ζ65 ζ6 linear of order 6 ρ4 1 1 1 1 -1 1 ζ3 ζ32 ζ32 ζ3 -1 1 1 1 1 1 ζ6 ζ65 linear of order 6 ρ5 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 linear of order 3 ρ7 2 2 2 2 0 -1 2 2 -1 -1 0 -1 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 2 2 2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 -1 -1 -1 -1 -1 0 0 complex lifted from C3×S3 ρ9 2 2 2 2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 -1 -1 -1 -1 -1 0 0 complex lifted from C3×S3 ρ10 3 3 -1 -1 -3 3 0 0 0 0 1 -1 -1 -1 -1 3 0 0 orthogonal lifted from C2×A4 ρ11 3 3 -1 -1 3 3 0 0 0 0 -1 -1 -1 -1 -1 3 0 0 orthogonal lifted from A4 ρ12 6 -2 -2 2 0 6 0 0 0 0 0 2 2 -2 -2 -2 0 0 orthogonal lifted from C24⋊C6 ρ13 6 6 -2 -2 0 -3 0 0 0 0 0 1 1 1 1 -3 0 0 orthogonal lifted from S3×A4 ρ14 6 -2 2 -2 0 6 0 0 0 0 0 -2 -2 2 2 -2 0 0 orthogonal lifted from C24⋊C6 ρ15 6 -2 2 -2 0 -3 0 0 0 0 0 1 1 -1-2√-3 -1+2√-3 1 0 0 complex faithful ρ16 6 -2 2 -2 0 -3 0 0 0 0 0 1 1 -1+2√-3 -1-2√-3 1 0 0 complex faithful ρ17 6 -2 -2 2 0 -3 0 0 0 0 0 -1+2√-3 -1-2√-3 1 1 1 0 0 complex faithful ρ18 6 -2 -2 2 0 -3 0 0 0 0 0 -1-2√-3 -1+2√-3 1 1 1 0 0 complex faithful

Permutation representations of (C22×S3)⋊A4
On 24 points - transitive group 24T696
Generators in S24
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 16)(2 18)(3 17)(4 14)(5 13)(6 15)(7 23)(8 22)(9 24)(10 20)(11 19)(12 21)
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 2 3)(4 11 9)(5 12 7)(6 10 8)(13 15 14)(16 24 20)(17 22 21)(18 23 19)

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

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

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

G:=TransitiveGroup(24,696);

On 24 points - transitive group 24T697
Generators in S24
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 16)(2 18)(3 17)(4 14)(5 13)(6 15)(7 23)(8 22)(9 24)(10 20)(11 19)(12 21)
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(4 10 7)(5 11 8)(6 12 9)(16 22 19)(17 23 20)(18 24 21)

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

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

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

G:=TransitiveGroup(24,697);

Matrix representation of (C22×S3)⋊A4 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 8 0 0 0 0 0 2 1 8 0 0 0 0 0 6 6 10 0 0 0 0 0 0 0 0 1 2 8 0 0 0 0 0 2 1 8 0 0 0 0 0 6 6 10
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 11 12 5 0 0 0 0 0 7 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 11 12 5 0 0 0 0 0 7 0 1
,
 3 0 0 0 0 0 0 0 0 9 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 11 12 5 0 0 0 0 0 7 0 1 0 0 12 0 0 0 0 0 0 0 11 12 5 0 0 0 0 0 7 0 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 8 0 0 0 0 0 2 1 8 0 0 0 0 0 6 6 10 0 0 0 0 0 0 0 0 12 11 5 0 0 0 0 0 0 12 0 0 0 0 0 0 0 7 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 11 5 0 0 0 0 0 0 12 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 11 12 5 0 0 0 0 0 7 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 12 12 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 5 0 0 0 0 0 0 0 1

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,2,6,0,0,0,0,0,2,1,6,0,0,0,0,0,8,8,10,0,0,0,0,0,0,0,0,1,2,6,0,0,0,0,0,2,1,6,0,0,0,0,0,8,8,10],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1],[3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,2,6,0,0,0,0,0,2,1,6,0,0,0,0,0,8,8,10,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,12,7,0,0,0,0,0,5,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,12,7,0,0,0,0,0,5,0,1,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,5,1] >;

(C22×S3)⋊A4 in GAP, Magma, Sage, TeX

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

G:=Group("(C2^2xS3):A4");
// GroupNames label

G:=SmallGroup(288,411);
// by ID

G=gap.SmallGroup(288,411);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-3,1640,198,1683,94,851,1524,9414]);
// Polycyclic

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

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