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G = C3×C42⋊S3order 288 = 25·32

Direct product of C3 and C42⋊S3

Aliases: C3×C42⋊S3, (C4×C12)⋊2S3, C42⋊C32C6, (C2×C6).1S4, C22.(C3×S4), C421(C3×S3), (C3×C42⋊C3)⋊6C2, SmallGroup(288,397)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊C3 — C3×C42⋊S3
 Chief series C1 — C22 — C42 — C42⋊C3 — C3×C42⋊C3 — C3×C42⋊S3
 Lower central C42⋊C3 — C3×C42⋊S3
 Upper central C1 — C3

Generators and relations for C3×C42⋊S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=c, dcd-1=b-1c-1, ece=b, ede=d-1 >

Character table of C3×C42⋊S3

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B 24C 24D size 1 3 12 1 1 32 32 32 3 3 6 12 3 3 12 12 12 12 3 3 3 3 6 6 12 12 12 12 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 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ5 1 1 -1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 -1 ζ32 ζ3 ζ6 ζ65 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ6 1 1 -1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 -1 ζ3 ζ32 ζ65 ζ6 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ7 2 2 0 2 2 -1 -1 -1 2 2 2 0 2 2 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 -1-√-3 -1+√-3 ζ6 ζ65 -1 2 2 2 0 -1-√-3 -1+√-3 0 0 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 0 0 0 0 0 0 complex lifted from C3×S3 ρ9 2 2 0 -1+√-3 -1-√-3 ζ65 ζ6 -1 2 2 2 0 -1+√-3 -1-√-3 0 0 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 0 0 0 0 0 0 complex lifted from C3×S3 ρ10 3 3 1 3 3 0 0 0 -1 -1 -1 1 3 3 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 orthogonal lifted from S4 ρ11 3 3 -1 3 3 0 0 0 -1 -1 -1 -1 3 3 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from S4 ρ12 3 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 1 1 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 ζ65 ζ6 ζ32 ζ32 ζ3 ζ3 complex lifted from C3×S4 ρ13 3 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 1 1 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 ζ6 ζ65 ζ3 ζ3 ζ32 ζ32 complex lifted from C3×S4 ρ14 3 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1 -1 -1 1 -3+3√-3/2 -3-3√-3/2 ζ3 ζ32 -1 -1 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×S4 ρ15 3 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1 -1 -1 1 -3-3√-3/2 -3+3√-3/2 ζ32 ζ3 -1 -1 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×S4 ρ16 3 -1 1 3 3 0 0 0 -1+2i -1-2i 1 -1 -1 -1 1 1 -i i -1+2i -1+2i -1-2i -1-2i 1 1 -1 -1 i -i i -i complex lifted from C42⋊S3 ρ17 3 -1 1 3 3 0 0 0 -1-2i -1+2i 1 -1 -1 -1 1 1 i -i -1-2i -1-2i -1+2i -1+2i 1 1 -1 -1 -i i -i i complex lifted from C42⋊S3 ρ18 3 -1 -1 3 3 0 0 0 -1+2i -1-2i 1 1 -1 -1 -1 -1 i -i -1+2i -1+2i -1-2i -1-2i 1 1 1 1 -i i -i i complex lifted from C42⋊S3 ρ19 3 -1 -1 3 3 0 0 0 -1-2i -1+2i 1 1 -1 -1 -1 -1 -i i -1-2i -1-2i -1+2i -1+2i 1 1 1 1 i -i i -i complex lifted from C42⋊S3 ρ20 3 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1-2i -1+2i 1 -1 ζ6 ζ65 ζ32 ζ3 i -i 2ζ43ζ3-ζ3 2ζ43ζ32-ζ32 2ζ4ζ3-ζ3 2ζ4ζ32-ζ32 ζ32 ζ3 ζ6 ζ65 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 complex faithful ρ21 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1+2i -1-2i 1 1 ζ65 ζ6 ζ65 ζ6 i -i 2ζ4ζ32-ζ32 2ζ4ζ3-ζ3 2ζ43ζ32-ζ32 2ζ43ζ3-ζ3 ζ3 ζ32 ζ3 ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 complex faithful ρ22 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1+2i -1-2i 1 1 ζ6 ζ65 ζ6 ζ65 i -i 2ζ4ζ3-ζ3 2ζ4ζ32-ζ32 2ζ43ζ3-ζ3 2ζ43ζ32-ζ32 ζ32 ζ3 ζ32 ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 complex faithful ρ23 3 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1-2i -1+2i 1 -1 ζ65 ζ6 ζ3 ζ32 i -i 2ζ43ζ32-ζ32 2ζ43ζ3-ζ3 2ζ4ζ32-ζ32 2ζ4ζ3-ζ3 ζ3 ζ32 ζ65 ζ6 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 complex faithful ρ24 3 -1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1+2i -1-2i 1 -1 ζ6 ζ65 ζ32 ζ3 -i i 2ζ4ζ3-ζ3 2ζ4ζ32-ζ32 2ζ43ζ3-ζ3 2ζ43ζ32-ζ32 ζ32 ζ3 ζ6 ζ65 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 complex faithful ρ25 3 -1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1+2i -1-2i 1 -1 ζ65 ζ6 ζ3 ζ32 -i i 2ζ4ζ32-ζ32 2ζ4ζ3-ζ3 2ζ43ζ32-ζ32 2ζ43ζ3-ζ3 ζ3 ζ32 ζ65 ζ6 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 complex faithful ρ26 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 -1-2i -1+2i 1 1 ζ65 ζ6 ζ65 ζ6 -i i 2ζ43ζ32-ζ32 2ζ43ζ3-ζ3 2ζ4ζ32-ζ32 2ζ4ζ3-ζ3 ζ3 ζ32 ζ3 ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 complex faithful ρ27 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 -1-2i -1+2i 1 1 ζ6 ζ65 ζ6 ζ65 -i i 2ζ43ζ3-ζ3 2ζ43ζ32-ζ32 2ζ4ζ3-ζ3 2ζ4ζ32-ζ32 ζ32 ζ3 ζ32 ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 complex faithful ρ28 6 -2 0 6 6 0 0 0 2 2 -2 0 -2 -2 0 0 0 0 2 2 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from C42⋊S3 ρ29 6 -2 0 -3+3√-3 -3-3√-3 0 0 0 2 2 -2 0 1-√-3 1+√-3 0 0 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 1-√-3 1+√-3 0 0 0 0 0 0 complex faithful ρ30 6 -2 0 -3-3√-3 -3+3√-3 0 0 0 2 2 -2 0 1+√-3 1-√-3 0 0 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 1+√-3 1-√-3 0 0 0 0 0 0 complex faithful

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

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

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

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

Matrix representation of C3×C42⋊S3 in GL3(𝔽13) generated by

 3 0 0 0 3 0 0 0 3
,
 5 0 0 0 5 0 0 0 12
,
 5 0 0 0 12 0 0 0 5
,
 0 0 9 9 0 0 0 9 0
,
 12 0 0 0 0 10 0 4 0
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[5,0,0,0,5,0,0,0,12],[5,0,0,0,12,0,0,0,5],[0,9,0,0,0,9,9,0,0],[12,0,0,0,0,4,0,10,0] >;

C3×C42⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC4^2:S3");
// GroupNames label

G:=SmallGroup(288,397);
// by ID

G=gap.SmallGroup(288,397);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,254,1011,185,360,634,1173,102,9077,1027,1784]);
// Polycyclic

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

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