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## G = S3×C42⋊C3order 288 = 25·32

### Direct product of S3 and C42⋊C3

Aliases: S3×C42⋊C3, (C4×C12)⋊4C6, (S3×C42)⋊C3, C424(C3×S3), C22.3(S3×A4), (C22×S3).3A4, C3⋊(C2×C42⋊C3), (C3×C42⋊C3)⋊5C2, (C2×C6).3(C2×A4), SmallGroup(288,407)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C12 — S3×C42⋊C3
 Chief series C1 — C22 — C2×C6 — C4×C12 — C3×C42⋊C3 — S3×C42⋊C3
 Lower central C4×C12 — S3×C42⋊C3
 Upper central C1

Generators and relations for S3×C42⋊C3
G = < a,b,c,d,e | a3=b2=c4=d4=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, ede-1=c-1d2 >

Subgroups: 354 in 59 conjugacy classes, 12 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, S3, C6, C2×C4, C23, C32, Dic3, C12, A4, D6, C2×C6, C42, C42, C22×C4, C3×S3, C4×S3, C2×Dic3, C2×C12, C2×A4, C22×S3, C2×C42, C3×A4, C42⋊C3, C42⋊C3, C4×Dic3, C4×C12, S3×C2×C4, S3×A4, C2×C42⋊C3, S3×C42, C3×C42⋊C3, S3×C42⋊C3
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, C42⋊C3, S3×A4, C2×C42⋊C3, S3×C42⋊C3

Character table of S3×C42⋊C3

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 12A 12B 12C 12D size 1 3 3 9 2 16 16 32 32 3 3 3 3 9 9 9 9 6 48 48 6 6 6 6 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 ζ3 ζ32 ζ32 ζ3 1 1 1 1 -1 -1 -1 -1 1 ζ6 ζ65 1 1 1 1 linear of order 6 ρ4 1 1 -1 -1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 -1 -1 -1 -1 1 ζ65 ζ6 1 1 1 1 linear of order 6 ρ5 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 linear of order 3 ρ6 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 linear of order 3 ρ7 2 2 0 0 -1 2 2 -1 -1 2 2 2 2 0 0 0 0 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 2 2 2 2 0 0 0 0 -1 0 0 -1 -1 -1 -1 complex lifted from C3×S3 ρ9 2 2 0 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 2 2 2 2 0 0 0 0 -1 0 0 -1 -1 -1 -1 complex lifted from C3×S3 ρ10 3 3 3 3 3 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 3 0 0 -1 -1 -1 -1 orthogonal lifted from A4 ρ11 3 3 -3 -3 3 0 0 0 0 -1 -1 -1 -1 1 1 1 1 3 0 0 -1 -1 -1 -1 orthogonal lifted from C2×A4 ρ12 3 -1 3 -1 3 0 0 0 0 -1+2i -1-2i 1 1 -1+2i -1-2i 1 1 -1 0 0 1 1 -1+2i -1-2i complex lifted from C42⋊C3 ρ13 3 -1 3 -1 3 0 0 0 0 1 1 -1+2i -1-2i 1 1 -1+2i -1-2i -1 0 0 -1+2i -1-2i 1 1 complex lifted from C42⋊C3 ρ14 3 -1 3 -1 3 0 0 0 0 -1-2i -1+2i 1 1 -1-2i -1+2i 1 1 -1 0 0 1 1 -1-2i -1+2i complex lifted from C42⋊C3 ρ15 3 -1 3 -1 3 0 0 0 0 1 1 -1-2i -1+2i 1 1 -1-2i -1+2i -1 0 0 -1-2i -1+2i 1 1 complex lifted from C42⋊C3 ρ16 3 -1 -3 1 3 0 0 0 0 -1-2i -1+2i 1 1 1+2i 1-2i -1 -1 -1 0 0 1 1 -1-2i -1+2i complex lifted from C2×C42⋊C3 ρ17 3 -1 -3 1 3 0 0 0 0 -1+2i -1-2i 1 1 1-2i 1+2i -1 -1 -1 0 0 1 1 -1+2i -1-2i complex lifted from C2×C42⋊C3 ρ18 3 -1 -3 1 3 0 0 0 0 1 1 -1+2i -1-2i -1 -1 1-2i 1+2i -1 0 0 -1+2i -1-2i 1 1 complex lifted from C2×C42⋊C3 ρ19 3 -1 -3 1 3 0 0 0 0 1 1 -1-2i -1+2i -1 -1 1+2i 1-2i -1 0 0 -1-2i -1+2i 1 1 complex lifted from C2×C42⋊C3 ρ20 6 6 0 0 -3 0 0 0 0 -2 -2 -2 -2 0 0 0 0 -3 0 0 1 1 1 1 orthogonal lifted from S3×A4 ρ21 6 -2 0 0 -3 0 0 0 0 -2+4i -2-4i 2 2 0 0 0 0 1 0 0 -1 -1 1-2i 1+2i complex faithful ρ22 6 -2 0 0 -3 0 0 0 0 2 2 -2+4i -2-4i 0 0 0 0 1 0 0 1-2i 1+2i -1 -1 complex faithful ρ23 6 -2 0 0 -3 0 0 0 0 2 2 -2-4i -2+4i 0 0 0 0 1 0 0 1+2i 1-2i -1 -1 complex faithful ρ24 6 -2 0 0 -3 0 0 0 0 -2-4i -2+4i 2 2 0 0 0 0 1 0 0 -1 -1 1+2i 1-2i complex faithful

Smallest permutation representation of S3×C42⋊C3
On 36 points
Generators in S36
(1 6 8)(2 7 11)(3 12 10)(4 5 9)(13 33 28)(14 34 25)(15 35 26)(16 36 27)(17 21 30)(18 22 31)(19 23 32)(20 24 29)
(1 6)(3 10)(4 9)(7 11)(13 33)(14 34)(15 35)(16 36)(17 30)(18 31)(19 32)(20 29)
(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 9 7 10)(2 12 8 5)(3 6 4 11)(13 14 15 16)(17 19)(18 20)(21 23)(22 24)(25 26 27 28)(29 31)(30 32)(33 34 35 36)
(1 29 14)(2 22 27)(3 17 33)(4 19 35)(5 23 26)(6 20 34)(7 31 16)(8 24 25)(9 32 15)(10 30 13)(11 18 36)(12 21 28)

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

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

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

Matrix representation of S3×C42⋊C3 in GL5(𝔽13)

 0 12 0 0 0 1 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 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 0 1 0 0 0 0 0 5 2 5 0 0 0 8 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 5 9 0 0 0 0 12 0 0 0 0 0 5
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 11 10 4

G:=sub<GL(5,GF(13))| [0,1,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,2,8,0,0,0,5,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,9,12,0,0,0,0,0,5],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,11,0,0,0,0,10,0,0,0,1,4] >;

S3×C42⋊C3 in GAP, Magma, Sage, TeX

S_3\times C_4^2\rtimes C_3
% in TeX

G:=Group("S3xC4^2:C3");
// GroupNames label

G:=SmallGroup(288,407);
// by ID

G=gap.SmallGroup(288,407);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-3,-2,2,198,772,2110,360,1684,3036,5305]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,e*d*e^-1=c^-1*d^2>;
// generators/relations

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