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## G = Q8.5S4order 192 = 26·3

### 3rd non-split extension by Q8 of S4 acting via S4/A4=C2

Aliases: Q8.5S4, U2(𝔽3)⋊2C2, SL2(𝔽3)⋊4D4, 2+ 1+41S3, C4.6(C2×S4), C4○D4.2D6, Q8.A41C2, C4.3S42C2, Q8.5(C3⋊D4), C4.A4.2C22, C2.13(A4⋊D4), SmallGroup(192,988)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C4.A4 — Q8.5S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — C4.3S4 — Q8.5S4
 Lower central SL2(𝔽3) — C4.A4 — Q8.5S4
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q8.5S4
G = < a,b,c,d,e,f | a4=e3=f2=1, b2=c2=d2=a2, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=a-1b, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >

Subgroups: 405 in 78 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×4], C22 [×7], S3, C6, C8 [×2], C2×C4 [×3], D4 [×10], Q8 [×2], C23 [×3], C12 [×2], D6, C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4 [×5], C4○D4, C4○D4 [×2], C3⋊C8, SL2(𝔽3), D12, C3×Q8, C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, Q82S3, GL2(𝔽3), C4.A4, C4.A4, D44D4, U2(𝔽3), C4.3S4, Q8.A4, Q8.5S4
Quotients: C1, C2 [×3], C22, S3, D4, D6, C3⋊D4, S4, C2×S4, A4⋊D4, Q8.5S4

Character table of Q8.5S4

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

Permutation representations of Q8.5S4
On 24 points - transitive group 24T401
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 10 3 12)(2 9 4 11)(5 21 7 23)(6 24 8 22)(13 17 15 19)(14 20 16 18)
(1 10 3 12)(2 11 4 9)(5 8 7 6)(13 18 15 20)(14 19 16 17)(21 22 23 24)
(1 4 3 2)(5 24 7 22)(6 21 8 23)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 17 7)(2 18 8)(3 19 5)(4 20 6)(9 14 22)(10 15 23)(11 16 24)(12 13 21)
(2 4)(5 19)(6 18)(7 17)(8 20)(9 12)(10 11)(13 22)(14 21)(15 24)(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), (1,10,3,12)(2,9,4,11)(5,21,7,23)(6,24,8,22)(13,17,15,19)(14,20,16,18), (1,10,3,12)(2,11,4,9)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,4,3,2)(5,24,7,22)(6,21,8,23)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,17,7)(2,18,8)(3,19,5)(4,20,6)(9,14,22)(10,15,23)(11,16,24)(12,13,21), (2,4)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,22)(14,21)(15,24)(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), (1,10,3,12)(2,9,4,11)(5,21,7,23)(6,24,8,22)(13,17,15,19)(14,20,16,18), (1,10,3,12)(2,11,4,9)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,4,3,2)(5,24,7,22)(6,21,8,23)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,17,7)(2,18,8)(3,19,5)(4,20,6)(9,14,22)(10,15,23)(11,16,24)(12,13,21), (2,4)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,22)(14,21)(15,24)(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)], [(1,10,3,12),(2,9,4,11),(5,21,7,23),(6,24,8,22),(13,17,15,19),(14,20,16,18)], [(1,10,3,12),(2,11,4,9),(5,8,7,6),(13,18,15,20),(14,19,16,17),(21,22,23,24)], [(1,4,3,2),(5,24,7,22),(6,21,8,23),(9,10,11,12),(13,17,15,19),(14,18,16,20)], [(1,17,7),(2,18,8),(3,19,5),(4,20,6),(9,14,22),(10,15,23),(11,16,24),(12,13,21)], [(2,4),(5,19),(6,18),(7,17),(8,20),(9,12),(10,11),(13,22),(14,21),(15,24),(16,23)])`

`G:=TransitiveGroup(24,401);`

Matrix representation of Q8.5S4 in GL4(ℚ) generated by

 0 0 0 1 0 0 1 0 0 -1 0 0 -1 0 0 0
,
 0 0 1 0 0 0 0 -1 -1 0 0 0 0 1 0 0
,
 0 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 0
,
 0 -1 0 0 1 0 0 0 0 0 0 1 0 0 -1 0
,
 -1/2 1/2 -1/2 1/2 -1/2 -1/2 -1/2 -1/2 1/2 1/2 -1/2 -1/2 -1/2 1/2 1/2 -1/2
,
 -1/2 1/2 -1/2 1/2 1/2 1/2 -1/2 -1/2 -1/2 -1/2 -1/2 -1/2 1/2 -1/2 -1/2 1/2
`G:=sub<GL(4,Rationals())| [0,0,0,-1,0,0,-1,0,0,1,0,0,1,0,0,0],[0,0,-1,0,0,0,0,1,1,0,0,0,0,-1,0,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[0,1,0,0,-1,0,0,0,0,0,0,-1,0,0,1,0],[-1/2,-1/2,1/2,-1/2,1/2,-1/2,1/2,1/2,-1/2,-1/2,-1/2,1/2,1/2,-1/2,-1/2,-1/2],[-1/2,1/2,-1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,1/2,-1/2,-1/2,1/2] >;`

Q8.5S4 in GAP, Magma, Sage, TeX

`Q_8._5S_4`
`% in TeX`

`G:=Group("Q8.5S4");`
`// GroupNames label`

`G:=SmallGroup(192,988);`
`// by ID`

`G=gap.SmallGroup(192,988);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,680,2102,1059,520,451,1684,655,172,1013,404,285,124]);`
`// Polycyclic`

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