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## G = D6.2S4order 288 = 25·32

### 2nd non-split extension by D6 of S4 acting via S4/A4=C2

Aliases: D6.2S4, CSU2(𝔽3)⋊2S3, SL2(𝔽3).6D6, Q8.7S32, C6.7(C2×S4), (S3×Q8)⋊2S3, C2.10(S3×S4), (C3×Q8).7D6, C6.6S43C2, C31(Q8.D6), (S3×SL2(𝔽3))⋊2C2, (C3×CSU2(𝔽3))⋊4C2, (C3×SL2(𝔽3)).6C22, SmallGroup(288,850)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — D6.2S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) — D6.2S4
 Lower central C3×SL2(𝔽3) — D6.2S4
 Upper central C1 — C2

Generators and relations for D6.2S4
G = < a,b,c,d,e,f | a6=b2=e3=1, c2=d2=f2=a3, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=a3b, dcd-1=a3c, ece-1=a3cd, fcf-1=cd, ede-1=c, fdf-1=a3d, fef-1=e-1 >

Subgroups: 558 in 85 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⋊S3, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, D12, C3⋊D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, S3×C6, C2×C3⋊S3, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), S3×Q8, Q83S3, C3⋊D12, C3×SL2(𝔽3), Q16⋊S3, Q8.D6, C3×CSU2(𝔽3), C6.6S4, S3×SL2(𝔽3), D6.2S4
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, Q8.D6, S3×S4, D6.2S4

Character table of D6.2S4

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 12A 12B 24A 24B size 1 1 6 36 2 8 16 6 12 18 2 8 16 24 24 12 36 12 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 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 0 2 2 -1 -1 -1 -1 0 0 2 0 0 0 orthogonal lifted from S3 ρ6 2 2 0 0 -1 2 -1 2 -2 0 -1 2 -1 0 0 -2 0 -1 1 1 1 orthogonal lifted from D6 ρ7 2 2 -2 0 2 -1 -1 2 0 -2 2 -1 -1 1 1 0 0 2 0 0 0 orthogonal lifted from D6 ρ8 2 2 0 0 -1 2 -1 2 2 0 -1 2 -1 0 0 2 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ9 3 3 3 1 3 0 0 -1 1 -1 3 0 0 0 0 -1 -1 -1 1 -1 -1 orthogonal lifted from S4 ρ10 3 3 3 -1 3 0 0 -1 -1 -1 3 0 0 0 0 1 1 -1 -1 1 1 orthogonal lifted from S4 ρ11 3 3 -3 -1 3 0 0 -1 1 1 3 0 0 0 0 -1 1 -1 1 -1 -1 orthogonal lifted from C2×S4 ρ12 3 3 -3 1 3 0 0 -1 -1 1 3 0 0 0 0 1 -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 -2 0 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 4 1 1 0 0 0 -4 -1 -1 √-3 -√-3 0 0 0 0 0 0 complex lifted from Q8.D6 ρ16 4 -4 0 0 4 1 1 0 0 0 -4 -1 -1 -√-3 √-3 0 0 0 0 0 0 complex lifted from Q8.D6 ρ17 4 -4 0 0 -2 -2 1 0 0 0 2 2 -1 0 0 0 0 0 0 -√-6 √-6 complex faithful ρ18 4 -4 0 0 -2 -2 1 0 0 0 2 2 -1 0 0 0 0 0 0 √-6 -√-6 complex faithful ρ19 6 6 0 0 -3 0 0 -2 2 0 -3 0 0 0 0 -2 0 1 -1 1 1 orthogonal lifted from S3×S4 ρ20 6 6 0 0 -3 0 0 -2 -2 0 -3 0 0 0 0 2 0 1 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 orthogonal faithful

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

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

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

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

Matrix representation of D6.2S4 in GL4(𝔽5) generated by

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

D6.2S4 in GAP, Magma, Sage, TeX

`D_6._2S_4`
`% in TeX`

`G:=Group("D6.2S4");`
`// GroupNames label`

`G:=SmallGroup(288,850);`
`// by ID`

`G=gap.SmallGroup(288,850);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,1016,93,675,1271,1908,172,768,1153,285,124]);`
`// Polycyclic`

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

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