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G = D6⋊S4order 288 = 25·32

1st semidirect product of D6 and S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×A4 — D6⋊S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C2×S3×A4 — D6⋊S4
 Lower central C3×A4 — C6×A4 — D6⋊S4
 Upper central C1 — C2

Generators and relations for D6⋊S4
G = < a,b,c,d,e,f | a6=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a3b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 790 in 130 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×5], C3, C3 [×2], C4 [×3], C22, C22 [×12], S3 [×3], C6, C6 [×6], C2×C4 [×3], D4 [×6], C23, C23 [×5], C32, Dic3 [×4], C12, A4, A4, D6, D6 [×7], C2×C6, C2×C6 [×6], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3 [×2], C3×C6, C2×Dic3 [×2], C3⋊D4 [×5], C2×C12, C3×D4 [×2], S4, C2×A4, C2×A4 [×2], C22×S3 [×4], C22×C6, C22×C6, C22≀C2, C3⋊Dic3, C3×A4, S3×C6 [×2], D6⋊C4 [×2], C6.D4, A4⋊C4 [×2], C2×C3⋊D4 [×2], C6×D4, C2×S4, C22×A4, S3×C23, D6⋊S3, C3×S4, S3×A4, C6×A4, C232D6, A4⋊D4, C6.7S4, C6×S4, C2×S3×A4, D6⋊S4
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], C3⋊D4 [×2], S4, S32, C2×S4, D6⋊S3, A4⋊D4, S3×S4, D6⋊S4

Character table of D6⋊S4

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B size 1 1 3 3 6 12 18 2 8 16 12 36 36 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 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 -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 linear of order 2 ρ5 2 2 2 2 0 2 0 -1 2 -1 2 0 0 -1 -1 -1 2 -1 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 2 0 2 2 -1 -1 0 0 0 2 2 2 -1 0 0 -1 -1 -1 0 0 orthogonal lifted from S3 ρ7 2 -2 -2 2 0 0 0 2 2 2 0 0 0 -2 2 -2 -2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ8 2 2 2 2 0 -2 0 -1 2 -1 -2 0 0 -1 -1 -1 2 1 1 -1 0 0 1 1 orthogonal lifted from D6 ρ9 2 2 2 2 -2 0 -2 2 -1 -1 0 0 0 2 2 2 -1 0 0 -1 1 1 0 0 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 0 -1 2 -1 0 0 0 1 -1 1 -2 √-3 -√-3 1 0 0 -√-3 √-3 complex lifted from C3⋊D4 ρ11 2 -2 -2 2 0 0 0 2 -1 -1 0 0 0 -2 2 -2 1 0 0 1 -√-3 √-3 0 0 complex lifted from C3⋊D4 ρ12 2 -2 -2 2 0 0 0 2 -1 -1 0 0 0 -2 2 -2 1 0 0 1 √-3 -√-3 0 0 complex lifted from C3⋊D4 ρ13 2 -2 -2 2 0 0 0 -1 2 -1 0 0 0 1 -1 1 -2 -√-3 √-3 1 0 0 √-3 -√-3 complex lifted from C3⋊D4 ρ14 3 3 -1 -1 3 -1 -1 3 0 0 1 1 -1 3 -1 -1 0 -1 -1 0 0 0 1 1 orthogonal lifted from S4 ρ15 3 3 -1 -1 -3 1 1 3 0 0 -1 1 -1 3 -1 -1 0 1 1 0 0 0 -1 -1 orthogonal lifted from C2×S4 ρ16 3 3 -1 -1 3 1 -1 3 0 0 -1 -1 1 3 -1 -1 0 1 1 0 0 0 -1 -1 orthogonal lifted from S4 ρ17 3 3 -1 -1 -3 -1 1 3 0 0 1 -1 1 3 -1 -1 0 -1 -1 0 0 0 1 1 orthogonal lifted from C2×S4 ρ18 4 4 4 4 0 0 0 -2 -2 1 0 0 0 -2 -2 -2 -2 0 0 1 0 0 0 0 orthogonal lifted from S32 ρ19 4 -4 -4 4 0 0 0 -2 -2 1 0 0 0 2 -2 2 2 0 0 -1 0 0 0 0 symplectic lifted from D6⋊S3, Schur index 2 ρ20 6 6 -2 -2 0 2 0 -3 0 0 -2 0 0 -3 1 1 0 -1 -1 0 0 0 1 1 orthogonal lifted from S3×S4 ρ21 6 6 -2 -2 0 -2 0 -3 0 0 2 0 0 -3 1 1 0 1 1 0 0 0 -1 -1 orthogonal lifted from S3×S4 ρ22 6 -6 2 -2 0 0 0 6 0 0 0 0 0 -6 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ23 6 -6 2 -2 0 0 0 -3 0 0 0 0 0 3 1 -1 0 -√-3 √-3 0 0 0 -√-3 √-3 complex faithful ρ24 6 -6 2 -2 0 0 0 -3 0 0 0 0 0 3 1 -1 0 √-3 -√-3 0 0 0 √-3 -√-3 complex faithful

Smallest permutation representation of D6⋊S4
On 36 points
Generators in S36
```(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)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 21)(22 24)(25 27)(28 30)(31 33)(34 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)```

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

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

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

Matrix representation of D6⋊S4 in GL5(𝔽13)

 4 0 0 0 0 6 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 3 3 0 0 0 6 10 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 12 12 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 12 12 0 0 0 0 1 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 12 12 12
,
 1 0 0 0 0 11 12 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 12 0

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

D6⋊S4 in GAP, Magma, Sage, TeX

`D_6\rtimes S_4`
`% in TeX`

`G:=Group("D6:S4");`
`// GroupNames label`

`G:=SmallGroup(288,857);`
`// by ID`

`G=gap.SmallGroup(288,857);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,234,1684,3036,782,1777,1350]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^6=b^2=c^2=d^2=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^3*b,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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