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## G = C2×S3×S4order 288 = 25·32

### Direct product of C2, S3 and S4

Aliases: C2×S3×S4, C23⋊S32, C61(C2×S4), C3⋊S4⋊C22, (C22×C6)⋊D6, (C6×S4)⋊1C2, (C2×A4)⋊1D6, (C3×A4)⋊C23, (S3×A4)⋊C22, (C6×A4)⋊C22, (C3×S4)⋊C22, (C22×S3)⋊D6, C31(C22×S4), A41(C22×S3), (S3×C23)⋊3S3, C22⋊(C2×S32), (C2×S3×A4)⋊5C2, (C2×C3⋊S4)⋊5C2, (C2×C6)⋊(C22×S3), Aut(S3×SL2(𝔽3)), SmallGroup(288,1028)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×S3×S4
G = < a,b,c,d,e,f,g | a2=b3=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, 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: 1594 in 272 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, A4, A4, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, S4, S4, C2×A4, C2×A4, C22×S3, C22×S3, C22×C6, C22×C6, C22×D4, S32, C3×A4, S3×C6, C2×C3⋊S3, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, C2×S4, C2×S4, C22×A4, S3×C23, S3×C23, C3×S4, C3⋊S4, S3×A4, C2×S32, C6×A4, C2×S3×D4, C22×S4, S3×S4, C6×S4, C2×C3⋊S4, C2×S3×A4, C2×S3×S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, S32, C2×S4, C2×S32, C22×S4, S3×S4, C2×S3×S4

Character table of C2×S3×S4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B size 1 1 3 3 3 3 6 6 9 9 18 18 2 8 16 6 6 18 18 2 6 6 8 12 12 16 24 24 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 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 -2 -2 -2 2 2 0 0 2 -2 0 0 2 -1 -1 0 0 0 0 -2 2 -2 1 0 0 1 1 -1 0 0 orthogonal lifted from D6 ρ10 2 2 2 2 2 2 0 0 2 2 0 0 2 -1 -1 0 0 0 0 2 2 2 -1 0 0 -1 -1 -1 0 0 orthogonal lifted from S3 ρ11 2 -2 0 -2 2 0 2 -2 0 0 0 0 -1 2 -1 2 -2 0 0 1 -1 1 -2 -1 1 1 0 0 1 -1 orthogonal lifted from D6 ρ12 2 2 0 2 2 0 2 2 0 0 0 0 -1 2 -1 2 2 0 0 -1 -1 -1 2 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ13 2 -2 2 -2 2 -2 0 0 -2 2 0 0 2 -1 -1 0 0 0 0 -2 2 -2 1 0 0 1 -1 1 0 0 orthogonal lifted from D6 ρ14 2 2 -2 2 2 -2 0 0 -2 -2 0 0 2 -1 -1 0 0 0 0 2 2 2 -1 0 0 -1 1 1 0 0 orthogonal lifted from D6 ρ15 2 -2 0 -2 2 0 -2 2 0 0 0 0 -1 2 -1 -2 2 0 0 1 -1 1 -2 1 -1 1 0 0 -1 1 orthogonal lifted from D6 ρ16 2 2 0 2 2 0 -2 -2 0 0 0 0 -1 2 -1 -2 -2 0 0 -1 -1 -1 2 1 1 -1 0 0 1 1 orthogonal lifted from D6 ρ17 3 -3 -3 1 -1 3 -1 1 -1 1 -1 1 3 0 0 1 -1 1 -1 -3 -1 1 0 -1 1 0 0 0 -1 1 orthogonal lifted from C2×S4 ρ18 3 3 -3 -1 -1 -3 1 1 1 1 -1 -1 3 0 0 -1 -1 1 1 3 -1 -1 0 1 1 0 0 0 -1 -1 orthogonal lifted from C2×S4 ρ19 3 3 3 -1 -1 3 -1 -1 -1 -1 -1 -1 3 0 0 1 1 1 1 3 -1 -1 0 -1 -1 0 0 0 1 1 orthogonal lifted from S4 ρ20 3 -3 3 1 -1 -3 1 -1 1 -1 -1 1 3 0 0 -1 1 1 -1 -3 -1 1 0 1 -1 0 0 0 1 -1 orthogonal lifted from C2×S4 ρ21 3 -3 -3 1 -1 3 1 -1 -1 1 1 -1 3 0 0 -1 1 -1 1 -3 -1 1 0 1 -1 0 0 0 1 -1 orthogonal lifted from C2×S4 ρ22 3 3 -3 -1 -1 -3 -1 -1 1 1 1 1 3 0 0 1 1 -1 -1 3 -1 -1 0 -1 -1 0 0 0 1 1 orthogonal lifted from C2×S4 ρ23 3 3 3 -1 -1 3 1 1 -1 -1 1 1 3 0 0 -1 -1 -1 -1 3 -1 -1 0 1 1 0 0 0 -1 -1 orthogonal lifted from S4 ρ24 3 -3 3 1 -1 -3 -1 1 1 -1 1 -1 3 0 0 1 -1 -1 1 -3 -1 1 0 -1 1 0 0 0 -1 1 orthogonal lifted from C2×S4 ρ25 4 -4 0 -4 4 0 0 0 0 0 0 0 -2 -2 1 0 0 0 0 2 -2 2 2 0 0 -1 0 0 0 0 orthogonal lifted from C2×S32 ρ26 4 4 0 4 4 0 0 0 0 0 0 0 -2 -2 1 0 0 0 0 -2 -2 -2 -2 0 0 1 0 0 0 0 orthogonal lifted from S32 ρ27 6 6 0 -2 -2 0 2 2 0 0 0 0 -3 0 0 -2 -2 0 0 -3 1 1 0 -1 -1 0 0 0 1 1 orthogonal lifted from S3×S4 ρ28 6 -6 0 2 -2 0 2 -2 0 0 0 0 -3 0 0 -2 2 0 0 3 1 -1 0 -1 1 0 0 0 -1 1 orthogonal faithful ρ29 6 -6 0 2 -2 0 -2 2 0 0 0 0 -3 0 0 2 -2 0 0 3 1 -1 0 1 -1 0 0 0 1 -1 orthogonal faithful ρ30 6 6 0 -2 -2 0 -2 -2 0 0 0 0 -3 0 0 2 2 0 0 -3 1 1 0 1 1 0 0 0 -1 -1 orthogonal lifted from S3×S4

Permutation representations of C2×S3×S4
On 18 points - transitive group 18T111
Generators in S18
(1 9)(2 7)(3 8)(4 18)(5 16)(6 17)(10 14)(11 15)(12 13)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(2 3)(4 5)(7 8)(10 12)(13 14)(16 18)
(1 9)(2 7)(3 8)(10 14)(11 15)(12 13)
(4 18)(5 16)(6 17)(10 14)(11 15)(12 13)
(1 11 17)(2 12 18)(3 10 16)(4 7 13)(5 8 14)(6 9 15)
(4 13)(5 14)(6 15)(10 16)(11 17)(12 18)

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

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

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

G:=TransitiveGroup(18,111);

On 24 points - transitive group 24T679
Generators in S24
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)
(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 7)(2 9)(3 8)(4 20)(5 19)(6 21)(10 24)(11 23)(12 22)(13 17)(14 16)(15 18)
(1 14)(2 15)(3 13)(4 24)(5 22)(6 23)(7 16)(8 17)(9 18)(10 20)(11 21)(12 19)
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(4 24 17)(5 22 18)(6 23 16)(10 13 20)(11 14 21)(12 15 19)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 18)(11 16)(12 17)(13 22)(14 23)(15 24)

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

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

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

G:=TransitiveGroup(24,679);

Matrix representation of C2×S3×S4 in GL5(ℤ)

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

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,1,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,-1,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,-1,0] >;

C2×S3×S4 in GAP, Magma, Sage, TeX

C_2\times S_3\times S_4
% in TeX

G:=Group("C2xS3xS4");
// GroupNames label

G:=SmallGroup(288,1028);
// by ID

G=gap.SmallGroup(288,1028);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,234,1684,3036,782,1777,1350]);
// Polycyclic

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