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## G = D4⋊S4order 192 = 26·3

### The semidirect product of D4 and S4 acting via S4/A4=C2

Aliases: D4⋊S4, A42D8, C4⋊S42C2, A4⋊C82C2, (D4×A4)⋊1C2, C4.2(C2×S4), C22⋊(D4⋊S3), (C2×A4).8D4, (C22×D4)⋊1S3, (C22×C4).2D6, (C4×A4).2C22, C2.5(A4⋊D4), C23.18(C3⋊D4), SmallGroup(192,974)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4⋊S4
G = < a,b,c,d,e,f | a4=b2=c2=d2=e3=f2=1, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 454 in 87 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×2], C22, C22 [×11], S3, C6 [×2], C8 [×2], C2×C4 [×3], D4, D4 [×8], C23, C23 [×5], C12, A4, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×5], C24, C3⋊C8, D12, C3×D4, S4, C2×A4, C2×A4, C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8 [×2], C22×D4, D4⋊S3, C4×A4, C2×S4, C22×A4, C22⋊D8, A4⋊C8, C4⋊S4, D4×A4, D4⋊S4
Quotients: C1, C2 [×3], C22, S3, D4, D6, D8, C3⋊D4, S4, D4⋊S3, C2×S4, A4⋊D4, D4⋊S4

Character table of D4⋊S4

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 6A 6B 6C 8A 8B 8C 8D 12 size 1 1 3 3 4 12 24 8 2 6 24 8 16 16 12 12 12 12 16 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 2 2 0 0 0 2 -2 -2 0 2 0 0 0 0 0 0 -2 orthogonal lifted from D4 ρ6 2 2 2 2 2 2 0 -1 2 2 0 -1 -1 -1 0 0 0 0 -1 orthogonal lifted from S3 ρ7 2 2 2 2 -2 -2 0 -1 2 2 0 -1 1 1 0 0 0 0 -1 orthogonal lifted from D6 ρ8 2 -2 2 -2 0 0 0 2 0 0 0 -2 0 0 √2 √2 -√2 -√2 0 orthogonal lifted from D8 ρ9 2 -2 2 -2 0 0 0 2 0 0 0 -2 0 0 -√2 -√2 √2 √2 0 orthogonal lifted from D8 ρ10 2 2 2 2 0 0 0 -1 -2 -2 0 -1 √-3 -√-3 0 0 0 0 1 complex lifted from C3⋊D4 ρ11 2 2 2 2 0 0 0 -1 -2 -2 0 -1 -√-3 √-3 0 0 0 0 1 complex lifted from C3⋊D4 ρ12 3 3 -1 -1 -3 1 -1 0 3 -1 1 0 0 0 1 -1 1 -1 0 orthogonal lifted from C2×S4 ρ13 3 3 -1 -1 3 -1 1 0 3 -1 -1 0 0 0 1 -1 1 -1 0 orthogonal lifted from S4 ρ14 3 3 -1 -1 3 -1 -1 0 3 -1 1 0 0 0 -1 1 -1 1 0 orthogonal lifted from S4 ρ15 3 3 -1 -1 -3 1 1 0 3 -1 -1 0 0 0 -1 1 -1 1 0 orthogonal lifted from C2×S4 ρ16 4 -4 4 -4 0 0 0 -2 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ17 6 6 -2 -2 0 0 0 0 -6 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4 ρ18 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 0 orthogonal faithful ρ19 6 -6 -2 2 0 0 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 0 orthogonal faithful

Permutation representations of D4⋊S4
On 24 points - transitive group 24T327
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 24)(2 23)(3 22)(4 21)(5 15)(6 14)(7 13)(8 16)(9 18)(10 17)(11 20)(12 19)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 9 16)(2 10 13)(3 11 14)(4 12 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(2 4)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(21 22)(23 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), (1,24)(2,23)(3,22)(4,21)(5,15)(6,14)(7,13)(8,16)(9,18)(10,17)(11,20)(12,19), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,9,16)(2,10,13)(3,11,14)(4,12,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (2,4)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(21,22)(23,24)>;`

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

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

`G:=TransitiveGroup(24,327);`

On 24 points - transitive group 24T330
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 7)(9 12)(10 11)(13 14)(15 16)(17 19)(21 23)
(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 16)(2 12 13)(3 9 14)(4 10 15)(5 19 21)(6 20 22)(7 17 23)(8 18 24)
(1 21)(2 24)(3 23)(4 22)(5 16)(6 15)(7 14)(8 13)(9 17)(10 20)(11 19)(12 18)```

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

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

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

`G:=TransitiveGroup(24,330);`

Matrix representation of D4⋊S4 in GL5(𝔽73)

 33 32 0 0 0 7 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 72 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 72 0 0 0 1 72 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 72 0 0 1 0 72 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 1 0 0 0 72 0 0 0 0 72 0 1
,
 1 0 0 0 0 39 72 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(5,GF(73))| [33,7,0,0,0,32,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,72,72,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,1,0,0,0,0,0,0,1],[1,39,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;`

D4⋊S4 in GAP, Magma, Sage, TeX

`D_4\rtimes S_4`
`% in TeX`

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

`G:=SmallGroup(192,974);`
`// by ID`

`G=gap.SmallGroup(192,974);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,85,254,135,58,1124,4037,285,2358,475]);`
`// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a*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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