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## G = (C2×C6)⋊S4order 288 = 25·32

### 2nd semidirect product of C2×C6 and S4 acting via S4/C22=S3

Aliases: (C2×C6)⋊2S4, C3⋊(C22⋊S4), C22⋊A44S3, (C23×C6)⋊6S3, C222(C3⋊S4), C244(C3⋊S3), (C3×C22⋊A4)⋊4C2, SmallGroup(288,1036)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×C22⋊A4 — (C2×C6)⋊S4
 Chief series C1 — C22 — C24 — C23×C6 — C3×C22⋊A4 — (C2×C6)⋊S4
 Lower central C3×C22⋊A4 — (C2×C6)⋊S4
 Upper central C1

Generators and relations for (C2×C6)⋊S4
G = < a,b,c,d,e,f | a2=b6=c2=d2=e3=f2=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=faf=b3, bc=cb, bd=db, fbf=ab2, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 1040 in 134 conjugacy classes, 15 normal (6 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, S4, C22×S3, C22×C6, C22≀C2, C3×A4, C6.D4, C2×C3⋊D4, C22⋊A4, C23×C6, C3⋊S4, C244S3, C22⋊S4, C3×C22⋊A4, (C2×C6)⋊S4
Quotients: C1, C2, S3, C3⋊S3, S4, C3⋊S4, C22⋊S4, (C2×C6)⋊S4

Character table of (C2×C6)⋊S4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 6E size 1 3 3 3 6 36 2 32 32 32 36 36 36 6 6 6 6 6 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 2 2 2 2 2 0 2 -1 -1 -1 0 0 0 2 2 2 2 2 orthogonal lifted from S3 ρ4 2 2 2 2 2 0 -1 -1 -1 2 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ5 2 2 2 2 2 0 -1 -1 2 -1 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 2 0 -1 2 -1 -1 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 3 -1 3 -1 -1 1 3 0 0 0 -1 1 -1 -1 -1 -1 -1 3 orthogonal lifted from S4 ρ8 3 -1 3 -1 -1 -1 3 0 0 0 1 -1 1 -1 -1 -1 -1 3 orthogonal lifted from S4 ρ9 3 -1 -1 3 -1 -1 3 0 0 0 -1 1 1 -1 3 -1 -1 -1 orthogonal lifted from S4 ρ10 3 3 -1 -1 -1 -1 3 0 0 0 1 1 -1 -1 -1 -1 3 -1 orthogonal lifted from S4 ρ11 3 -1 -1 3 -1 1 3 0 0 0 1 -1 -1 -1 3 -1 -1 -1 orthogonal lifted from S4 ρ12 3 3 -1 -1 -1 1 3 0 0 0 -1 -1 1 -1 -1 -1 3 -1 orthogonal lifted from S4 ρ13 6 -2 -2 -2 2 0 6 0 0 0 0 0 0 2 -2 2 -2 -2 orthogonal lifted from C22⋊S4 ρ14 6 6 -2 -2 -2 0 -3 0 0 0 0 0 0 1 1 1 -3 1 orthogonal lifted from C3⋊S4 ρ15 6 -2 -2 6 -2 0 -3 0 0 0 0 0 0 1 -3 1 1 1 orthogonal lifted from C3⋊S4 ρ16 6 -2 6 -2 -2 0 -3 0 0 0 0 0 0 1 1 1 1 -3 orthogonal lifted from C3⋊S4 ρ17 6 -2 -2 -2 2 0 -3 0 0 0 0 0 0 -1-2√-3 1 -1+2√-3 1 1 complex faithful ρ18 6 -2 -2 -2 2 0 -3 0 0 0 0 0 0 -1+2√-3 1 -1-2√-3 1 1 complex faithful

Permutation representations of (C2×C6)⋊S4
On 24 points - transitive group 24T699
Generators in S24
(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 19)(8 20)(9 21)(10 22)(11 23)(12 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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)
(1 4)(2 5)(3 6)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 16)(14 17)(15 18)
(2 14 17)(4 16 13)(6 18 15)(7 19 10)(8 11 23)(9 21 12)
(1 22)(2 12)(3 20)(4 10)(5 24)(6 8)(7 13)(9 17)(11 15)(14 21)(16 19)(18 23)

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

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

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

G:=TransitiveGroup(24,699);

Matrix representation of (C2×C6)⋊S4 in GL8(ℤ)

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

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

(C2×C6)⋊S4 in GAP, Magma, Sage, TeX

(C_2\times C_6)\rtimes S_4
% in TeX

G:=Group("(C2xC6):S4");
// GroupNames label

G:=SmallGroup(288,1036);
// by ID

G=gap.SmallGroup(288,1036);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,57,254,1011,514,634,956,6053,4548,3534,1777]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^6=c^2=d^2=e^3=f^2=1,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=f*a*f=b^3,b*c=c*b,b*d=d*b,f*b*f=a*b^2,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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