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## G = C42⋊C3⋊S3order 288 = 25·32

### 1st semidirect product of C42⋊C3 and S3 acting via S3/C3=C2

Aliases: (C4×C12)⋊1C6, C42⋊C31S3, C423S3⋊C3, C422(C3×S3), C3⋊(C42⋊C6), C22.2(S3×A4), (C22×S3).2A4, (C3×C42⋊C3)⋊1C2, (C2×C6).2(C2×A4), SmallGroup(288,406)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C42⋊C3⋊S3
G = < a,b,c,d,e | a4=b4=c3=d3=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=ab2, cbc-1=a-1b2, bd=db, ebe=a2b-1, cd=dc, ce=ec, ede=d-1 >

Character table of C42⋊C3⋊S3

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 12A 12B 12C 12D size 1 3 12 2 16 16 32 32 6 6 36 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 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 ζ32 ζ3 ζ32 ζ3 1 1 -1 1 ζ65 ζ6 1 1 1 1 linear of order 6 ρ4 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 ζ32 ζ3 1 1 1 1 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 ζ3 ζ32 1 1 1 1 linear of order 3 ρ6 1 1 -1 1 ζ3 ζ32 ζ3 ζ32 1 1 -1 1 ζ6 ζ65 1 1 1 1 linear of order 6 ρ7 2 2 0 -1 2 2 -1 -1 2 2 0 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 2 2 0 -1 0 0 -1 -1 -1 -1 complex lifted from C3×S3 ρ9 2 2 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 2 2 0 -1 0 0 -1 -1 -1 -1 complex lifted from C3×S3 ρ10 3 3 3 3 0 0 0 0 -1 -1 -1 3 0 0 -1 -1 -1 -1 orthogonal lifted from A4 ρ11 3 3 -3 3 0 0 0 0 -1 -1 1 3 0 0 -1 -1 -1 -1 orthogonal lifted from C2×A4 ρ12 6 6 0 -3 0 0 0 0 -2 -2 0 -3 0 0 1 1 1 1 orthogonal lifted from S3×A4 ρ13 6 -2 0 6 0 0 0 0 2i -2i 0 -2 0 0 2i -2i -2i 2i complex lifted from C42⋊C6 ρ14 6 -2 0 6 0 0 0 0 -2i 2i 0 -2 0 0 -2i 2i 2i -2i complex lifted from C42⋊C6 ρ15 6 -2 0 -3 0 0 0 0 2i -2i 0 1 0 0 2ζ4ζ32-2ζ32-1 2ζ43ζ3-2ζ3-1 2ζ43ζ32-2ζ32-1 2ζ4ζ3-2ζ3-1 complex faithful ρ16 6 -2 0 -3 0 0 0 0 2i -2i 0 1 0 0 2ζ4ζ3-2ζ3-1 2ζ43ζ32-2ζ32-1 2ζ43ζ3-2ζ3-1 2ζ4ζ32-2ζ32-1 complex faithful ρ17 6 -2 0 -3 0 0 0 0 -2i 2i 0 1 0 0 2ζ43ζ3-2ζ3-1 2ζ4ζ32-2ζ32-1 2ζ4ζ3-2ζ3-1 2ζ43ζ32-2ζ32-1 complex faithful ρ18 6 -2 0 -3 0 0 0 0 -2i 2i 0 1 0 0 2ζ43ζ32-2ζ32-1 2ζ4ζ3-2ζ3-1 2ζ4ζ32-2ζ32-1 2ζ43ζ3-2ζ3-1 complex faithful

Smallest permutation representation of C42⋊C3⋊S3
On 48 points
Generators in S48
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 36 6 40)(2 33 7 37)(3 34 8 38)(4 35 5 39)(9 43 25 16)(10 44 26 13)(11 41 27 14)(12 42 28 15)(17 29 21 47)(18 30 22 48)(19 31 23 45)(20 32 24 46)
(2 34 37)(3 6 8)(4 38 35)(5 36 33)(7 40 39)(9 13 14)(10 12 28)(11 44 43)(15 41 27)(16 25 42)(17 21 19)(18 45 46)(20 31 30)(22 29 48)(24 47 32)
(1 23 26)(2 24 27)(3 21 28)(4 22 25)(5 18 9)(6 19 10)(7 20 11)(8 17 12)(13 36 45)(14 33 46)(15 34 47)(16 35 48)(29 42 38)(30 43 39)(31 44 40)(32 41 37)
(2 7)(4 5)(9 22)(10 19)(11 24)(12 17)(13 29)(14 48)(15 31)(16 46)(18 25)(20 27)(21 28)(23 26)(30 41)(32 43)(33 35)(34 40)(36 38)(37 39)(42 45)(44 47)```

`G:=sub<Sym(48)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,36,6,40)(2,33,7,37)(3,34,8,38)(4,35,5,39)(9,43,25,16)(10,44,26,13)(11,41,27,14)(12,42,28,15)(17,29,21,47)(18,30,22,48)(19,31,23,45)(20,32,24,46), (2,34,37)(3,6,8)(4,38,35)(5,36,33)(7,40,39)(9,13,14)(10,12,28)(11,44,43)(15,41,27)(16,25,42)(17,21,19)(18,45,46)(20,31,30)(22,29,48)(24,47,32), (1,23,26)(2,24,27)(3,21,28)(4,22,25)(5,18,9)(6,19,10)(7,20,11)(8,17,12)(13,36,45)(14,33,46)(15,34,47)(16,35,48)(29,42,38)(30,43,39)(31,44,40)(32,41,37), (2,7)(4,5)(9,22)(10,19)(11,24)(12,17)(13,29)(14,48)(15,31)(16,46)(18,25)(20,27)(21,28)(23,26)(30,41)(32,43)(33,35)(34,40)(36,38)(37,39)(42,45)(44,47)>;`

`G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,36,6,40)(2,33,7,37)(3,34,8,38)(4,35,5,39)(9,43,25,16)(10,44,26,13)(11,41,27,14)(12,42,28,15)(17,29,21,47)(18,30,22,48)(19,31,23,45)(20,32,24,46), (2,34,37)(3,6,8)(4,38,35)(5,36,33)(7,40,39)(9,13,14)(10,12,28)(11,44,43)(15,41,27)(16,25,42)(17,21,19)(18,45,46)(20,31,30)(22,29,48)(24,47,32), (1,23,26)(2,24,27)(3,21,28)(4,22,25)(5,18,9)(6,19,10)(7,20,11)(8,17,12)(13,36,45)(14,33,46)(15,34,47)(16,35,48)(29,42,38)(30,43,39)(31,44,40)(32,41,37), (2,7)(4,5)(9,22)(10,19)(11,24)(12,17)(13,29)(14,48)(15,31)(16,46)(18,25)(20,27)(21,28)(23,26)(30,41)(32,43)(33,35)(34,40)(36,38)(37,39)(42,45)(44,47) );`

`G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,36,6,40),(2,33,7,37),(3,34,8,38),(4,35,5,39),(9,43,25,16),(10,44,26,13),(11,41,27,14),(12,42,28,15),(17,29,21,47),(18,30,22,48),(19,31,23,45),(20,32,24,46)], [(2,34,37),(3,6,8),(4,38,35),(5,36,33),(7,40,39),(9,13,14),(10,12,28),(11,44,43),(15,41,27),(16,25,42),(17,21,19),(18,45,46),(20,31,30),(22,29,48),(24,47,32)], [(1,23,26),(2,24,27),(3,21,28),(4,22,25),(5,18,9),(6,19,10),(7,20,11),(8,17,12),(13,36,45),(14,33,46),(15,34,47),(16,35,48),(29,42,38),(30,43,39),(31,44,40),(32,41,37)], [(2,7),(4,5),(9,22),(10,19),(11,24),(12,17),(13,29),(14,48),(15,31),(16,46),(18,25),(20,27),(21,28),(23,26),(30,41),(32,43),(33,35),(34,40),(36,38),(37,39),(42,45),(44,47)]])`

Matrix representation of C42⋊C3⋊S3 in GL6(𝔽13)

 5 7 8 4 7 11 12 4 7 0 7 7 1 8 6 4 0 4 1 0 8 12 11 5 12 7 0 10 1 6 5 4 6 7 9 8
,
 6 9 11 6 0 6 5 7 12 9 9 0 3 8 5 2 9 6 6 7 5 8 6 7 0 8 12 5 1 12 3 0 11 10 3 2
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 12 12 12
,
 12 0 0 1 12 0 0 12 0 1 0 12 12 12 11 2 1 1 9 9 9 1 0 0 10 9 9 1 0 0 9 10 9 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 4 4 4 12 0 0 3 4 4 0 12 0 4 3 4 0 0 12

`G:=sub<GL(6,GF(13))| [5,12,1,1,12,5,7,4,8,0,7,4,8,7,6,8,0,6,4,0,4,12,10,7,7,7,0,11,1,9,11,7,4,5,6,8],[6,5,3,6,0,3,9,7,8,7,8,0,11,12,5,5,12,11,6,9,2,8,5,10,0,9,9,6,1,3,6,0,6,7,12,2],[0,0,1,0,0,1,1,0,0,0,0,1,0,1,0,0,0,1,0,0,0,1,0,12,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,12,9,10,9,0,12,12,9,9,10,0,0,11,9,9,9,1,1,2,1,1,1,12,0,1,0,0,0,0,12,1,0,0,0],[1,0,0,4,3,4,0,1,0,4,4,3,0,0,1,4,4,4,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;`

C42⋊C3⋊S3 in GAP, Magma, Sage, TeX

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

`G:=Group("C4^2:C3:S3");`
`// GroupNames label`

`G:=SmallGroup(288,406);`
`// by ID`

`G=gap.SmallGroup(288,406);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,2,-3,-2,2,4664,198,520,4371,1102,192,1684,3036,5305]);`
`// Polycyclic`

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

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