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## G = D6.S4order 288 = 25·32

### 1st non-split extension by D6 of S4 acting via S4/A4=C2

Aliases: D6.1S4, GL2(𝔽3)⋊2S3, SL2(𝔽3).5D6, Q8.6S32, C2.9(S3×S4), C6.6(C2×S4), (S3×Q8)⋊1S3, (C3×Q8).6D6, C6.5S44C2, C32(Q8.D6), (S3×SL2(𝔽3))⋊1C2, (C3×GL2(𝔽3))⋊2C2, (C3×SL2(𝔽3)).5C22, SmallGroup(288,849)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — D6.S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) — D6.S4
 Lower central C3×SL2(𝔽3) — D6.S4
 Upper central C1 — C2

Generators and relations for D6.S4
G = < a,b,c,d,e,f | a6=b2=e3=f2=1, c2=d2=a3, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a3b, dcd-1=a3c, ece-1=a3cd, fcf=cd, ede-1=c, fdf=a3d, fef=e-1 >

Subgroups: 470 in 83 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, D6, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C8.C22, C3⋊Dic3, S3×C6, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), D42S3, S3×Q8, D6⋊S3, C3×SL2(𝔽3), D4.D6, Q8.D6, C3×GL2(𝔽3), C6.5S4, S3×SL2(𝔽3), D6.S4
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, Q8.D6, S3×S4, D6.S4

Character table of D6.S4

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 6F 8A 8B 12 24A 24B size 1 1 6 12 2 8 16 6 18 36 2 8 16 24 24 24 12 36 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 trivial ρ2 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 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 linear of order 2 ρ5 2 2 -2 0 2 -1 -1 2 -2 0 2 -1 -1 0 1 1 0 0 2 0 0 orthogonal lifted from D6 ρ6 2 2 0 2 -1 2 -1 2 0 0 -1 2 -1 -1 0 0 2 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 0 2 -1 -1 2 2 0 2 -1 -1 0 -1 -1 0 0 2 0 0 orthogonal lifted from S3 ρ8 2 2 0 -2 -1 2 -1 2 0 0 -1 2 -1 1 0 0 -2 0 -1 1 1 orthogonal lifted from D6 ρ9 3 3 3 1 3 0 0 -1 -1 1 3 0 0 1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S4 ρ10 3 3 -3 1 3 0 0 -1 1 -1 3 0 0 1 0 0 -1 1 -1 -1 -1 orthogonal lifted from C2×S4 ρ11 3 3 3 -1 3 0 0 -1 -1 -1 3 0 0 -1 0 0 1 1 -1 1 1 orthogonal lifted from S4 ρ12 3 3 -3 -1 3 0 0 -1 1 1 3 0 0 -1 0 0 1 -1 -1 1 1 orthogonal lifted from C2×S4 ρ13 4 4 0 0 -2 -2 1 4 0 0 -2 -2 1 0 0 0 0 0 -2 0 0 orthogonal lifted from S32 ρ14 4 -4 0 0 4 -2 -2 0 0 0 -4 2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8.D6, Schur index 2 ρ15 4 -4 0 0 -2 -2 1 0 0 0 2 2 -1 0 0 0 0 0 0 -√6 √6 symplectic faithful, Schur index 2 ρ16 4 -4 0 0 -2 -2 1 0 0 0 2 2 -1 0 0 0 0 0 0 √6 -√6 symplectic faithful, Schur index 2 ρ17 4 -4 0 0 4 1 1 0 0 0 -4 -1 -1 0 √-3 -√-3 0 0 0 0 0 complex lifted from Q8.D6 ρ18 4 -4 0 0 4 1 1 0 0 0 -4 -1 -1 0 -√-3 √-3 0 0 0 0 0 complex lifted from Q8.D6 ρ19 6 6 0 -2 -3 0 0 -2 0 0 -3 0 0 1 0 0 2 0 1 -1 -1 orthogonal lifted from S3×S4 ρ20 6 6 0 2 -3 0 0 -2 0 0 -3 0 0 -1 0 0 -2 0 1 1 1 orthogonal lifted from S3×S4 ρ21 8 -8 0 0 -4 2 -1 0 0 0 4 -2 1 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D6.S4
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 6)(2 5)(3 4)(7 11)(8 10)(13 15)(16 18)(19 20)(21 24)(22 23)(25 26)(27 30)(28 29)(31 33)(34 36)(37 42)(38 41)(39 40)(43 47)(44 46)
(1 26 4 29)(2 27 5 30)(3 28 6 25)(7 33 10 36)(8 34 11 31)(9 35 12 32)(13 44 16 47)(14 45 17 48)(15 46 18 43)(19 39 22 42)(20 40 23 37)(21 41 24 38)
(1 40 4 37)(2 41 5 38)(3 42 6 39)(7 15 10 18)(8 16 11 13)(9 17 12 14)(19 28 22 25)(20 29 23 26)(21 30 24 27)(31 47 34 44)(32 48 35 45)(33 43 36 46)
(7 33 43)(8 34 44)(9 35 45)(10 36 46)(11 31 47)(12 32 48)(19 28 42)(20 29 37)(21 30 38)(22 25 39)(23 26 40)(24 27 41)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)```

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

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

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

Matrix representation of D6.S4 in GL4(𝔽5) generated by

 3 4 4 1 0 2 4 4 1 2 4 1 4 1 0 3
,
 3 1 3 2 3 0 3 3 0 0 4 0 2 1 3 3
,
 4 2 1 1 1 0 4 1 0 0 2 0 1 2 1 4
,
 1 4 0 3 3 2 3 0 2 1 3 2 2 1 1 4
,
 0 1 2 4 0 4 0 2 2 1 4 2 0 2 0 0
,
 1 0 2 2 0 4 0 0 0 3 4 3 0 2 0 1
`G:=sub<GL(4,GF(5))| [3,0,1,4,4,2,2,1,4,4,4,0,1,4,1,3],[3,3,0,2,1,0,0,1,3,3,4,3,2,3,0,3],[4,1,0,1,2,0,0,2,1,4,2,1,1,1,0,4],[1,3,2,2,4,2,1,1,0,3,3,1,3,0,2,4],[0,0,2,0,1,4,1,2,2,0,4,0,4,2,2,0],[1,0,0,0,0,4,3,2,2,0,4,0,2,0,3,1] >;`

D6.S4 in GAP, Magma, Sage, TeX

`D_6.S_4`
`% in TeX`

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

`G:=SmallGroup(288,849);`
`// by ID`

`G=gap.SmallGroup(288,849);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,93,675,1271,1908,172,768,1153,285,124]);`
`// Polycyclic`

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

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