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## G = D4⋊2S4order 192 = 26·3

### The semidirect product of D4 and S4 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — D4⋊2S4
 Chief series C1 — C22 — A4 — C2×A4 — C2×S4 — C4×S4 — D4⋊2S4
 Lower central A4 — C2×A4 — D4⋊2S4
 Upper central C1 — C2 — D4

Generators and relations for D42S4
G = < a,b,c,d,e,f | a4=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a2b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 654 in 173 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×8], C22, C22 [×2], C22 [×15], S3, C6 [×3], C2×C4 [×17], D4, D4 [×12], Q8 [×2], C23, C23 [×9], Dic3 [×3], C12, A4, D6, C2×C6 [×2], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×5], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24 [×2], Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, S4, C2×A4, C2×A4 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, A4⋊C4, A4⋊C4 [×2], C4×A4, D42S3, C2×S4, C22×A4 [×2], D45D4, A4⋊Q8, C4×S4, C2×A4⋊C4 [×2], A4⋊D4 [×2], D4×A4, D42S4
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4, S4, C22×S3, D42S3, C2×S4 [×3], C22×S4, D42S4

Character table of D42S4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 6A 6B 6C 12 size 1 1 2 2 3 3 6 6 12 8 2 6 6 6 6 6 12 12 12 12 12 8 16 16 16 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 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 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 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 -1 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 -2 2 2 2 -2 0 -1 -2 -2 0 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 2 2 -2 -2 0 -1 2 2 0 0 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 2 2 2 2 0 -1 2 2 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 2 2 2 -2 2 0 -1 -2 -2 0 0 0 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ13 2 -2 0 0 -2 2 0 0 0 2 0 0 2i 2i -2i -2i 0 0 0 0 0 -2 0 0 0 complex lifted from C4○D4 ρ14 2 -2 0 0 -2 2 0 0 0 2 0 0 -2i -2i 2i 2i 0 0 0 0 0 -2 0 0 0 complex lifted from C4○D4 ρ15 3 3 -3 3 -1 -1 1 -1 -1 0 -3 1 1 -1 1 -1 -1 1 1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 3 -3 -1 -1 -1 1 -1 0 -3 1 1 -1 1 -1 1 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ17 3 3 -3 -3 -1 -1 1 1 -1 0 3 -1 -1 1 -1 1 -1 -1 1 1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ18 3 3 3 3 -1 -1 -1 -1 -1 0 3 -1 -1 1 -1 1 1 1 1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ19 3 3 3 3 -1 -1 -1 -1 1 0 3 -1 1 -1 1 -1 -1 -1 -1 1 1 0 0 0 0 orthogonal lifted from S4 ρ20 3 3 -3 -3 -1 -1 1 1 1 0 3 -1 1 -1 1 -1 1 1 -1 -1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ21 3 3 3 -3 -1 -1 -1 1 1 0 -3 1 -1 1 -1 1 -1 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ22 3 3 -3 3 -1 -1 1 -1 1 0 -3 1 -1 1 -1 1 1 -1 -1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ23 4 -4 0 0 -4 4 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ24 6 -6 0 0 2 -2 0 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 complex faithful ρ25 6 -6 0 0 2 -2 0 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of D42S4
On 24 points - transitive group 24T295
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 22)(23 24)
(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(1 11 13)(2 12 14)(3 9 15)(4 10 16)(5 18 21)(6 19 22)(7 20 23)(8 17 24)
(1 23)(2 24)(3 21)(4 22)(5 15)(6 16)(7 13)(8 14)(9 18)(10 19)(11 20)(12 17)```

`G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (1,11,13)(2,12,14)(3,9,15)(4,10,16)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,23)(2,24)(3,21)(4,22)(5,15)(6,16)(7,13)(8,14)(9,18)(10,19)(11,20)(12,17)>;`

`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), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (1,11,13)(2,12,14)(3,9,15)(4,10,16)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,23)(2,24)(3,21)(4,22)(5,15)(6,16)(7,13)(8,14)(9,18)(10,19)(11,20)(12,17) );`

`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)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,22),(23,24)], [(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(13,15),(14,16),(21,23),(22,24)], [(1,11,13),(2,12,14),(3,9,15),(4,10,16),(5,18,21),(6,19,22),(7,20,23),(8,17,24)], [(1,23),(2,24),(3,21),(4,22),(5,15),(6,16),(7,13),(8,14),(9,18),(10,19),(11,20),(12,17)])`

`G:=TransitiveGroup(24,295);`

On 24 points - transitive group 24T318
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 22)(2 21)(3 24)(4 23)(5 11)(6 10)(7 9)(8 12)(13 18)(14 17)(15 20)(16 19)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 11 19)(2 12 20)(3 9 17)(4 10 18)(5 16 22)(6 13 23)(7 14 24)(8 15 21)
(5 14)(6 15)(7 16)(8 13)(9 17)(10 18)(11 19)(12 20)(21 23)(22 24)```

`G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22)(2,21)(3,24)(4,23)(5,11)(6,10)(7,9)(8,12)(13,18)(14,17)(15,20)(16,19), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,11,19)(2,12,20)(3,9,17)(4,10,18)(5,16,22)(6,13,23)(7,14,24)(8,15,21), (5,14)(6,15)(7,16)(8,13)(9,17)(10,18)(11,19)(12,20)(21,23)(22,24)>;`

`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), (1,22)(2,21)(3,24)(4,23)(5,11)(6,10)(7,9)(8,12)(13,18)(14,17)(15,20)(16,19), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,11,19)(2,12,20)(3,9,17)(4,10,18)(5,16,22)(6,13,23)(7,14,24)(8,15,21), (5,14)(6,15)(7,16)(8,13)(9,17)(10,18)(11,19)(12,20)(21,23)(22,24) );`

`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)], [(1,22),(2,21),(3,24),(4,23),(5,11),(6,10),(7,9),(8,12),(13,18),(14,17),(15,20),(16,19)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(21,23),(22,24)], [(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,11,19),(2,12,20),(3,9,17),(4,10,18),(5,16,22),(6,13,23),(7,14,24),(8,15,21)], [(5,14),(6,15),(7,16),(8,13),(9,17),(10,18),(11,19),(12,20),(21,23),(22,24)])`

`G:=TransitiveGroup(24,318);`

On 24 points - transitive group 24T359
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(9 11)(13 15)(18 20)(22 24)
(1 6)(2 7)(3 8)(4 5)(9 11)(10 12)(13 15)(14 16)(17 23)(18 24)(19 21)(20 22)
(1 3)(2 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 21)(18 22)(19 23)(20 24)
(1 21 14)(2 22 15)(3 23 16)(4 24 13)(5 20 9)(6 17 10)(7 18 11)(8 19 12)
(1 7)(2 8)(3 5)(4 6)(9 23)(10 24)(11 21)(12 22)(13 17)(14 18)(15 19)(16 20)```

`G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(9,11)(13,15)(18,20)(22,24), (1,6)(2,7)(3,8)(4,5)(9,11)(10,12)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14)(17,21)(18,22)(19,23)(20,24), (1,21,14)(2,22,15)(3,23,16)(4,24,13)(5,20,9)(6,17,10)(7,18,11)(8,19,12), (1,7)(2,8)(3,5)(4,6)(9,23)(10,24)(11,21)(12,22)(13,17)(14,18)(15,19)(16,20)>;`

`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), (2,4)(5,7)(9,11)(13,15)(18,20)(22,24), (1,6)(2,7)(3,8)(4,5)(9,11)(10,12)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14)(17,21)(18,22)(19,23)(20,24), (1,21,14)(2,22,15)(3,23,16)(4,24,13)(5,20,9)(6,17,10)(7,18,11)(8,19,12), (1,7)(2,8)(3,5)(4,6)(9,23)(10,24)(11,21)(12,22)(13,17)(14,18)(15,19)(16,20) );`

`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)], [(2,4),(5,7),(9,11),(13,15),(18,20),(22,24)], [(1,6),(2,7),(3,8),(4,5),(9,11),(10,12),(13,15),(14,16),(17,23),(18,24),(19,21),(20,22)], [(1,3),(2,4),(5,7),(6,8),(9,15),(10,16),(11,13),(12,14),(17,21),(18,22),(19,23),(20,24)], [(1,21,14),(2,22,15),(3,23,16),(4,24,13),(5,20,9),(6,17,10),(7,18,11),(8,19,12)], [(1,7),(2,8),(3,5),(4,6),(9,23),(10,24),(11,21),(12,22),(13,17),(14,18),(15,19),(16,20)])`

`G:=TransitiveGroup(24,359);`

On 24 points - transitive group 24T396
Generators in S24
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 12)(2 11)(3 10)(4 9)(5 21)(6 24)(7 23)(8 22)(13 17)(14 20)(15 19)(16 18)
(1 12)(2 9)(3 10)(4 11)(5 23)(6 24)(7 21)(8 22)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 21)(6 22)(7 23)(8 24)(9 11)(10 12)(13 17)(14 18)(15 19)(16 20)
(1 5 18)(2 6 19)(3 7 20)(4 8 17)(9 22 13)(10 23 14)(11 24 15)(12 21 16)
(5 18)(6 19)(7 20)(8 17)(9 11)(10 12)(13 24)(14 21)(15 22)(16 23)```

`G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,21)(6,24)(7,23)(8,22)(13,17)(14,20)(15,19)(16,18), (1,12)(2,9)(3,10)(4,11)(5,23)(6,24)(7,21)(8,22)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,21)(6,22)(7,23)(8,24)(9,11)(10,12)(13,17)(14,18)(15,19)(16,20), (1,5,18)(2,6,19)(3,7,20)(4,8,17)(9,22,13)(10,23,14)(11,24,15)(12,21,16), (5,18)(6,19)(7,20)(8,17)(9,11)(10,12)(13,24)(14,21)(15,22)(16,23)>;`

`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), (1,12)(2,11)(3,10)(4,9)(5,21)(6,24)(7,23)(8,22)(13,17)(14,20)(15,19)(16,18), (1,12)(2,9)(3,10)(4,11)(5,23)(6,24)(7,21)(8,22)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,21)(6,22)(7,23)(8,24)(9,11)(10,12)(13,17)(14,18)(15,19)(16,20), (1,5,18)(2,6,19)(3,7,20)(4,8,17)(9,22,13)(10,23,14)(11,24,15)(12,21,16), (5,18)(6,19)(7,20)(8,17)(9,11)(10,12)(13,24)(14,21)(15,22)(16,23) );`

`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)], [(1,12),(2,11),(3,10),(4,9),(5,21),(6,24),(7,23),(8,22),(13,17),(14,20),(15,19),(16,18)], [(1,12),(2,9),(3,10),(4,11),(5,23),(6,24),(7,21),(8,22),(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(5,21),(6,22),(7,23),(8,24),(9,11),(10,12),(13,17),(14,18),(15,19),(16,20)], [(1,5,18),(2,6,19),(3,7,20),(4,8,17),(9,22,13),(10,23,14),(11,24,15),(12,21,16)], [(5,18),(6,19),(7,20),(8,17),(9,11),(10,12),(13,24),(14,21),(15,22),(16,23)])`

`G:=TransitiveGroup(24,396);`

Matrix representation of D42S4 in GL5(𝔽13)

 8 0 0 0 0 11 5 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 5 0 0 0 0 1 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 12 0 0 0 0 0 12 0 0 0 8 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 8 12 0 0 0 5 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 5 0 11 0 0 0 0 5 0 0 2 1 8
,
 1 0 0 0 0 3 12 0 0 0 0 0 1 0 0 0 0 8 0 1 0 0 5 1 0

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

D42S4 in GAP, Magma, Sage, TeX

`D_4\rtimes_2S_4`
`% in TeX`

`G:=Group("D4:2S4");`
`// GroupNames label`

`G:=SmallGroup(192,1473);`
`// by ID`

`G=gap.SmallGroup(192,1473);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,64,254,135,1124,4037,285,2358,475]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^4=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^2*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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