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## G = D4.5S4order 192 = 26·3

### 2nd 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) — D4.5S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C4.6S4 — D4.5S4
 Lower central SL2(𝔽3) — D4.5S4
 Upper central C1 — C2 — D4

Generators and relations for D4.5S4
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, fbf=a2b, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >

Subgroups: 471 in 140 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, D4, D4, Q8, Q8, Dic3, C12, D6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, 2- 1+4, CSU2(𝔽3), CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C4.A4, D42S3, Q8○D8, C2×CSU2(𝔽3), Q8.D6, C4.S4, C4.6S4, D4.A4, D4.5S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, C22×S4, D4.5S4

Character table of D4.5S4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 8C 8D 8E 12 size 1 1 2 2 6 12 8 2 6 6 6 12 12 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 -1 -2 2 2 -2 0 0 0 -1 1 -1 0 0 0 0 0 1 orthogonal lifted from D6 ρ10 2 2 2 -2 -2 0 -1 -2 -2 2 2 0 0 0 -1 -1 1 0 0 0 0 0 1 orthogonal lifted from D6 ρ11 2 2 -2 -2 2 0 -1 2 -2 2 -2 0 0 0 -1 1 1 0 0 0 0 0 -1 orthogonal lifted from D6 ρ12 2 2 2 2 2 0 -1 2 2 2 2 0 0 0 -1 -1 -1 0 0 0 0 0 -1 orthogonal lifted from S3 ρ13 3 3 -3 3 1 1 0 -3 -1 -1 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 0 3 1 -1 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 0 -3 1 -1 -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 0 3 -1 -1 -1 -1 -1 -1 0 0 0 1 1 1 1 1 0 orthogonal lifted from S4 ρ17 3 3 -3 3 1 -1 0 -3 -1 -1 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 0 3 1 -1 1 1 1 -1 0 0 0 1 1 -1 1 -1 0 orthogonal lifted from C2×S4 ρ19 3 3 3 -3 1 1 0 -3 1 -1 -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 0 3 -1 -1 -1 1 1 1 0 0 0 -1 -1 -1 -1 -1 0 orthogonal lifted from S4 ρ21 4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 2 0 0 -2√2 2√2 0 0 0 0 symplectic faithful, Schur index 2 ρ22 4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 2 0 0 2√2 -2√2 0 0 0 0 symplectic faithful, Schur index 2 ρ23 8 -8 0 0 0 0 2 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of D4.5S4
On 32 points
Generators in S32
```(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)
(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 26)(27 28)(29 30)(31 32)
(1 8 3 6)(2 5 4 7)(9 24 11 22)(10 21 12 23)(13 19 15 17)(14 20 16 18)(25 29 27 31)(26 30 28 32)
(1 19 3 17)(2 20 4 18)(5 14 7 16)(6 15 8 13)(9 25 11 27)(10 26 12 28)(21 32 23 30)(22 29 24 31)
(5 20 16)(6 17 13)(7 18 14)(8 19 15)(9 24 31)(10 21 32)(11 22 29)(12 23 30)
(1 25)(2 26)(3 27)(4 28)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 32)```

`G:=sub<Sym(32)| (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), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32), (1,8,3,6)(2,5,4,7)(9,24,11,22)(10,21,12,23)(13,19,15,17)(14,20,16,18)(25,29,27,31)(26,30,28,32), (1,19,3,17)(2,20,4,18)(5,14,7,16)(6,15,8,13)(9,25,11,27)(10,26,12,28)(21,32,23,30)(22,29,24,31), (5,20,16)(6,17,13)(7,18,14)(8,19,15)(9,24,31)(10,21,32)(11,22,29)(12,23,30), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32)>;`

`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), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32), (1,8,3,6)(2,5,4,7)(9,24,11,22)(10,21,12,23)(13,19,15,17)(14,20,16,18)(25,29,27,31)(26,30,28,32), (1,19,3,17)(2,20,4,18)(5,14,7,16)(6,15,8,13)(9,25,11,27)(10,26,12,28)(21,32,23,30)(22,29,24,31), (5,20,16)(6,17,13)(7,18,14)(8,19,15)(9,24,31)(10,21,32)(11,22,29)(12,23,30), (1,25)(2,26)(3,27)(4,28)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18)(13,29)(14,30)(15,31)(16,32) );`

`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)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,26),(27,28),(29,30),(31,32)], [(1,8,3,6),(2,5,4,7),(9,24,11,22),(10,21,12,23),(13,19,15,17),(14,20,16,18),(25,29,27,31),(26,30,28,32)], [(1,19,3,17),(2,20,4,18),(5,14,7,16),(6,15,8,13),(9,25,11,27),(10,26,12,28),(21,32,23,30),(22,29,24,31)], [(5,20,16),(6,17,13),(7,18,14),(8,19,15),(9,24,31),(10,21,32),(11,22,29),(12,23,30)], [(1,25),(2,26),(3,27),(4,28),(5,21),(6,22),(7,23),(8,24),(9,19),(10,20),(11,17),(12,18),(13,29),(14,30),(15,31),(16,32)]])`

Matrix representation of D4.5S4 in GL4(𝔽7) generated by

 6 2 0 5 5 0 4 3 2 5 4 3 6 6 4 4
,
 4 0 5 2 5 0 0 3 2 5 4 3 5 5 1 6
,
 6 6 1 1 2 0 4 1 3 3 0 1 4 3 5 1
,
 4 0 6 5 6 0 1 1 5 5 2 4 6 1 4 1
,
 1 1 4 1 0 3 3 3 6 6 0 2 1 6 3 1
,
 6 6 3 6 6 3 4 4 6 3 6 3 5 0 4 6
`G:=sub<GL(4,GF(7))| [6,5,2,6,2,0,5,6,0,4,4,4,5,3,3,4],[4,5,2,5,0,0,5,5,5,0,4,1,2,3,3,6],[6,2,3,4,6,0,3,3,1,4,0,5,1,1,1,1],[4,6,5,6,0,0,5,1,6,1,2,4,5,1,4,1],[1,0,6,1,1,3,6,6,4,3,0,3,1,3,2,1],[6,6,6,5,6,3,3,0,3,4,6,4,6,4,3,6] >;`

D4.5S4 in GAP, Magma, Sage, TeX

`D_4._5S_4`
`% in TeX`

`G:=Group("D4.5S4");`
`// GroupNames label`

`G:=SmallGroup(192,1486);`
`// by ID`

`G=gap.SmallGroup(192,1486);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,1059,451,1684,655,172,1013,404,285,124]);`
`// 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,f*b*f=a^2*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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