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## G = D4.4S4order 192 = 26·3

### 1st non-split extension by D4 of S4 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — D4.4S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C2×GL2(𝔽3) — D4.4S4
 Lower central SL2(𝔽3) — D4.4S4
 Upper central C1 — C2 — D4

Generators and relations for D4.4S4
G = < a,b,c,d,e,f | a4=b2=e3=f2=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >

Subgroups: 615 in 152 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×4], C22 [×2], C22 [×7], S3 [×4], C6 [×3], C8 [×4], C2×C4 [×7], D4, D4 [×12], Q8, Q8 [×4], C23 [×3], Dic3, C12, D6 [×7], C2×C6 [×2], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×2], C2×Q8, C4○D4, C4○D4 [×7], SL2(𝔽3), C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, CSU2(𝔽3), GL2(𝔽3), GL2(𝔽3) [×2], C2×SL2(𝔽3) [×2], C4.A4, S3×D4, D4○SD16, C2×GL2(𝔽3) [×2], Q8.D6 [×2], C4.6S4, C4.3S4, D4.A4, D4.4S4
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, C2×S4 [×3], C22×S4, D4.4S4

Character table of D4.4S4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 8D 8E 12 size 1 1 2 2 6 12 12 12 8 2 6 6 6 12 8 16 16 6 6 12 12 12 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 -2 -2 0 0 0 -1 -2 2 2 -2 0 -1 -1 1 0 0 0 0 0 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 2 0 0 0 -1 2 -2 2 -2 0 -1 1 1 0 0 0 0 0 -1 orthogonal lifted from D6 ρ11 2 2 -2 2 -2 0 0 0 -1 -2 -2 2 2 0 -1 1 -1 0 0 0 0 0 1 orthogonal lifted from D6 ρ12 2 2 2 2 2 0 0 0 -1 2 2 2 2 0 -1 -1 -1 0 0 0 0 0 -1 orthogonal lifted from S3 ρ13 3 3 -3 3 1 -1 1 1 0 -3 1 -1 -1 -1 0 0 0 -1 -1 1 -1 1 0 orthogonal lifted from C2×S4 ρ14 3 3 -3 -3 -1 1 -1 1 0 3 1 -1 1 -1 0 0 0 1 1 -1 -1 1 0 orthogonal lifted from C2×S4 ρ15 3 3 3 -3 1 1 1 -1 0 -3 -1 -1 1 -1 0 0 0 -1 -1 -1 1 1 0 orthogonal lifted from C2×S4 ρ16 3 3 3 3 -1 -1 -1 -1 0 3 -1 -1 -1 -1 0 0 0 1 1 1 1 1 0 orthogonal lifted from S4 ρ17 3 3 3 -3 1 -1 -1 1 0 -3 -1 -1 1 1 0 0 0 1 1 1 -1 -1 0 orthogonal lifted from C2×S4 ρ18 3 3 3 3 -1 1 1 1 0 3 -1 -1 -1 1 0 0 0 -1 -1 -1 -1 -1 0 orthogonal lifted from S4 ρ19 3 3 -3 3 1 1 -1 -1 0 -3 1 -1 -1 1 0 0 0 1 1 -1 1 -1 0 orthogonal lifted from C2×S4 ρ20 3 3 -3 -3 -1 -1 1 -1 0 3 1 -1 1 1 0 0 0 -1 -1 1 1 -1 0 orthogonal lifted from C2×S4 ρ21 4 -4 0 0 0 0 0 0 -2 0 0 0 0 0 2 0 0 2√-2 -2√-2 0 0 0 0 complex faithful ρ22 4 -4 0 0 0 0 0 0 -2 0 0 0 0 0 2 0 0 -2√-2 2√-2 0 0 0 0 complex faithful ρ23 8 -8 0 0 0 0 0 0 2 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D4.4S4
On 16 points - transitive group 16T422
Generators in S16
```(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(5 7)(9 11)(14 16)
(1 6 3 8)(2 7 4 5)(9 16 11 14)(10 13 12 15)
(1 10 3 12)(2 11 4 9)(5 14 7 16)(6 15 8 13)
(5 9 16)(6 10 13)(7 11 14)(8 12 15)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)```

`G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,7)(9,11)(14,16), (1,6,3,8)(2,7,4,5)(9,16,11,14)(10,13,12,15), (1,10,3,12)(2,11,4,9)(5,14,7,16)(6,15,8,13), (5,9,16)(6,10,13)(7,11,14)(8,12,15), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)>;`

`G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,7)(9,11)(14,16), (1,6,3,8)(2,7,4,5)(9,16,11,14)(10,13,12,15), (1,10,3,12)(2,11,4,9)(5,14,7,16)(6,15,8,13), (5,9,16)(6,10,13)(7,11,14)(8,12,15), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15) );`

`G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(2,4),(5,7),(9,11),(14,16)], [(1,6,3,8),(2,7,4,5),(9,16,11,14),(10,13,12,15)], [(1,10,3,12),(2,11,4,9),(5,14,7,16),(6,15,8,13)], [(5,9,16),(6,10,13),(7,11,14),(8,12,15)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15)])`

`G:=TransitiveGroup(16,422);`

Matrix representation of D4.4S4 in GL4(𝔽3) generated by

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

D4.4S4 in GAP, Magma, Sage, TeX

`D_4._4S_4`
`% in TeX`

`G:=Group("D4.4S4");`
`// GroupNames label`

`G:=SmallGroup(192,1485);`
`// by ID`

`G=gap.SmallGroup(192,1485);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,1059,451,1684,655,172,1013,404,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^4=b^2=e^3=f^2=1,c^2=d^2=a^2,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,b*f=f*b,d*c*d^-1=a^2*c,e*c*e^-1=a^2*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=a^2*d,f*e*f=e^-1>;`
`// generators/relations`

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