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## G = Q8.4S4order 192 = 26·3

### 2nd non-split extension by Q8 of S4 acting via S4/A4=C2

Aliases: Q8.4S4, U2(𝔽3)⋊1C2, SL2(𝔽3).8D4, 2+ 1+4.1S3, C4.5(C2×S4), Q8.A4.C2, C4○D4.1D6, C4.S42C2, Q8.4(C3⋊D4), C4.A4.1C22, C2.12(A4⋊D4), SmallGroup(192,987)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C4.A4 — Q8.4S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — C4.S4 — Q8.4S4
 Lower central SL2(𝔽3) — C4.A4 — Q8.4S4
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q8.4S4
G = < a,b,c,d,e,f | a4=e3=1, b2=c2=d2=f2=a2, bab-1=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=a-1b, dcd-1=a2c, ece-1=a2cd, fcf-1=cd, ede-1=c, fdf-1=a2d, fef-1=e-1 >

Subgroups: 293 in 69 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×4], C6, C8 [×2], C2×C4 [×4], D4 [×4], Q8 [×2], Q8 [×2], C23 [×2], Dic3, C12 [×2], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×2], C3⋊C8, SL2(𝔽3), Dic6, C3×Q8, C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, C3⋊Q16, CSU2(𝔽3), C4.A4, C4.A4, D4.9D4, U2(𝔽3), C4.S4, Q8.A4, Q8.4S4
Quotients: C1, C2 [×3], C22, S3, D4, D6, C3⋊D4, S4, C2×S4, A4⋊D4, Q8.4S4

Character table of Q8.4S4

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

Smallest permutation representation of Q8.4S4
On 48 points
Generators in S48
```(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 9 3 11)(2 12 4 10)(5 48 7 46)(6 47 8 45)(13 20 15 18)(14 19 16 17)(21 25 23 27)(22 28 24 26)(29 36 31 34)(30 35 32 33)(37 44 39 42)(38 43 40 41)
(1 2 3 4)(5 8 7 6)(9 12 11 10)(13 17 15 19)(14 18 16 20)(21 27 23 25)(22 28 24 26)(29 34 31 36)(30 35 32 33)(37 41 39 43)(38 42 40 44)(45 46 47 48)
(1 9 3 11)(2 10 4 12)(5 48 7 46)(6 45 8 47)(13 14 15 16)(17 20 19 18)(21 26 23 28)(22 27 24 25)(29 33 31 35)(30 34 32 36)(37 38 39 40)(41 44 43 42)
(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 43 35)(6 44 36)(7 41 33)(8 42 34)(9 17 26)(10 18 27)(11 19 28)(12 20 25)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(1 30 3 32)(2 29 4 31)(5 25 7 27)(6 28 8 26)(9 36 11 34)(10 35 12 33)(13 37 15 39)(14 40 16 38)(17 44 19 42)(18 43 20 41)(21 45 23 47)(22 48 24 46)```

`G:=sub<Sym(48)| (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,9,3,11)(2,12,4,10)(5,48,7,46)(6,47,8,45)(13,20,15,18)(14,19,16,17)(21,25,23,27)(22,28,24,26)(29,36,31,34)(30,35,32,33)(37,44,39,42)(38,43,40,41), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,17,15,19)(14,18,16,20)(21,27,23,25)(22,28,24,26)(29,34,31,36)(30,35,32,33)(37,41,39,43)(38,42,40,44)(45,46,47,48), (1,9,3,11)(2,10,4,12)(5,48,7,46)(6,45,8,47)(13,14,15,16)(17,20,19,18)(21,26,23,28)(22,27,24,25)(29,33,31,35)(30,34,32,36)(37,38,39,40)(41,44,43,42), (1,16,24)(2,13,21)(3,14,22)(4,15,23)(5,43,35)(6,44,36)(7,41,33)(8,42,34)(9,17,26)(10,18,27)(11,19,28)(12,20,25)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (1,30,3,32)(2,29,4,31)(5,25,7,27)(6,28,8,26)(9,36,11,34)(10,35,12,33)(13,37,15,39)(14,40,16,38)(17,44,19,42)(18,43,20,41)(21,45,23,47)(22,48,24,46)>;`

`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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,9,3,11)(2,12,4,10)(5,48,7,46)(6,47,8,45)(13,20,15,18)(14,19,16,17)(21,25,23,27)(22,28,24,26)(29,36,31,34)(30,35,32,33)(37,44,39,42)(38,43,40,41), (1,2,3,4)(5,8,7,6)(9,12,11,10)(13,17,15,19)(14,18,16,20)(21,27,23,25)(22,28,24,26)(29,34,31,36)(30,35,32,33)(37,41,39,43)(38,42,40,44)(45,46,47,48), (1,9,3,11)(2,10,4,12)(5,48,7,46)(6,45,8,47)(13,14,15,16)(17,20,19,18)(21,26,23,28)(22,27,24,25)(29,33,31,35)(30,34,32,36)(37,38,39,40)(41,44,43,42), (1,16,24)(2,13,21)(3,14,22)(4,15,23)(5,43,35)(6,44,36)(7,41,33)(8,42,34)(9,17,26)(10,18,27)(11,19,28)(12,20,25)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (1,30,3,32)(2,29,4,31)(5,25,7,27)(6,28,8,26)(9,36,11,34)(10,35,12,33)(13,37,15,39)(14,40,16,38)(17,44,19,42)(18,43,20,41)(21,45,23,47)(22,48,24,46) );`

`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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,9,3,11),(2,12,4,10),(5,48,7,46),(6,47,8,45),(13,20,15,18),(14,19,16,17),(21,25,23,27),(22,28,24,26),(29,36,31,34),(30,35,32,33),(37,44,39,42),(38,43,40,41)], [(1,2,3,4),(5,8,7,6),(9,12,11,10),(13,17,15,19),(14,18,16,20),(21,27,23,25),(22,28,24,26),(29,34,31,36),(30,35,32,33),(37,41,39,43),(38,42,40,44),(45,46,47,48)], [(1,9,3,11),(2,10,4,12),(5,48,7,46),(6,45,8,47),(13,14,15,16),(17,20,19,18),(21,26,23,28),(22,27,24,25),(29,33,31,35),(30,34,32,36),(37,38,39,40),(41,44,43,42)], [(1,16,24),(2,13,21),(3,14,22),(4,15,23),(5,43,35),(6,44,36),(7,41,33),(8,42,34),(9,17,26),(10,18,27),(11,19,28),(12,20,25),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(1,30,3,32),(2,29,4,31),(5,25,7,27),(6,28,8,26),(9,36,11,34),(10,35,12,33),(13,37,15,39),(14,40,16,38),(17,44,19,42),(18,43,20,41),(21,45,23,47),(22,48,24,46)])`

Matrix representation of Q8.4S4 in GL4(𝔽5) generated by

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

Q8.4S4 in GAP, Magma, Sage, TeX

`Q_8._4S_4`
`% in TeX`

`G:=Group("Q8.4S4");`
`// GroupNames label`

`G:=SmallGroup(192,987);`
`// by ID`

`G=gap.SmallGroup(192,987);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,680,2102,1059,520,451,1684,655,172,1013,404,285,124]);`
`// Polycyclic`

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

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