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## G = Q16⋊S3order 96 = 25·3

### 2nd semidirect product of Q16 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q16⋊S3
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×Q8 — Q16⋊S3
 Lower central C3 — C6 — C12 — Q16⋊S3
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16⋊S3
G = < a,b,c,d | a8=c3=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 146 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Q8, Dic3, Dic3, C12, C12, D6, D6, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C3×Q8, C8.C22, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, S3×Q8, Q83S3, Q16⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8.C22, S3×D4, Q16⋊S3

Character table of Q16⋊S3

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6 8A 8B 12A 12B 12C 24A 24B size 1 1 6 12 2 2 4 4 6 12 2 4 12 4 8 8 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 0 0 -1 2 2 2 0 0 -1 2 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 0 2 -2 0 0 2 0 2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 0 0 -1 2 2 -2 0 0 -1 -2 0 -1 -1 1 1 1 orthogonal lifted from D6 ρ12 2 2 0 0 -1 2 -2 2 0 0 -1 -2 0 -1 1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 0 0 -1 2 -2 -2 0 0 -1 2 0 -1 1 1 -1 -1 orthogonal lifted from D6 ρ14 2 2 2 0 2 -2 0 0 -2 0 2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ15 4 4 0 0 -2 -4 0 0 0 0 -2 0 0 2 0 0 0 0 orthogonal lifted from S3×D4 ρ16 4 -4 0 0 4 0 0 0 0 0 -4 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ17 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 -√-6 √-6 complex faithful ρ18 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 √-6 -√-6 complex faithful

Smallest permutation representation of Q16⋊S3
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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 39 13 35)(10 38 14 34)(11 37 15 33)(12 36 16 40)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)
(1 13 46)(2 14 47)(3 15 48)(4 16 41)(5 9 42)(6 10 43)(7 11 44)(8 12 45)(17 37 28)(18 38 29)(19 39 30)(20 40 31)(21 33 32)(22 34 25)(23 35 26)(24 36 27)
(1 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)(33 48)(34 45)(35 42)(36 47)(37 44)(38 41)(39 46)(40 43)```

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

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

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

Matrix representation of Q16⋊S3 in GL4(𝔽5) generated by

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

Q16⋊S3 in GAP, Magma, Sage, TeX

`Q_{16}\rtimes S_3`
`% in TeX`

`G:=Group("Q16:S3");`
`// GroupNames label`

`G:=SmallGroup(96,125);`
`// by ID`

`G=gap.SmallGroup(96,125);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,116,86,297,159,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^3=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;`
`// generators/relations`

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