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## G = Q16.A4order 192 = 26·3

### The non-split extension by Q16 of A4 acting through Inn(Q16)

Aliases: Q16.A4, SL2(𝔽3).11D4, 2+ 1+4.2C6, D4○D8⋊C3, C8○D4.C6, C8.A43C2, C8.2(C2×A4), C2.9(D4×A4), Q8.A44C2, Q8.3(C3×D4), Q8.4(C2×A4), C4.4(C22×A4), C4.A4.15C22, C4○D4.1(C2×C6), SmallGroup(192,1017)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4 — Q16.A4
 Chief series C1 — C2 — Q8 — C4○D4 — C4.A4 — Q8.A4 — Q16.A4
 Lower central Q8 — C4○D4 — Q16.A4
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Q16.A4
G = < a,b,c,d,e | a8=e3=1, b2=c2=d2=a4, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >

Subgroups: 291 in 73 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×5], C6, C8, C8, C2×C4 [×3], D4 [×7], Q8, Q8 [×2], C23 [×2], C12 [×3], C2×C8, M4(2), D8 [×3], SD16 [×2], Q16, C2×D4 [×4], C4○D4, C4○D4 [×4], C24, SL2(𝔽3), C3×Q8 [×2], C8○D4, C2×D8, C4○D8, C8⋊C22 [×2], 2+ 1+4 [×2], C3×Q16, C4.A4, C4.A4 [×2], D4○D8, C8.A4, Q8.A4 [×2], Q16.A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, C3×D4, C2×A4 [×3], C22×A4, D4×A4, Q16.A4

Character table of Q16.A4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 6A 6B 8A 8B 8C 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D size 1 1 6 12 12 4 4 2 4 4 6 4 4 2 2 12 8 8 16 16 16 16 8 8 8 8 ρ1 1 1 1 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 -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 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 -1 -1 -1 linear of order 2 ρ5 1 1 1 -1 -1 ζ3 ζ32 1 -1 -1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ6 1 1 1 -1 1 ζ32 ζ3 1 -1 1 1 ζ32 ζ3 -1 -1 -1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ7 1 1 1 1 -1 ζ3 ζ32 1 1 -1 1 ζ3 ζ32 -1 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ8 1 1 1 -1 -1 ζ32 ζ3 1 -1 -1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ9 1 1 1 -1 1 ζ3 ζ32 1 -1 1 1 ζ3 ζ32 -1 -1 -1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ10 1 1 1 1 -1 ζ32 ζ3 1 1 -1 1 ζ32 ζ3 -1 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ11 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ12 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ13 2 2 -2 0 0 2 2 -2 0 0 2 2 2 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 -2 0 0 -1+√-3 -1-√-3 -2 0 0 2 -1+√-3 -1-√-3 0 0 0 1-√-3 1+√-3 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ15 2 2 -2 0 0 -1-√-3 -1+√-3 -2 0 0 2 -1-√-3 -1+√-3 0 0 0 1+√-3 1-√-3 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ16 3 3 -1 1 -1 0 0 3 -3 3 -1 0 0 -3 -3 1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 3 3 -1 -1 -1 0 0 3 3 3 -1 0 0 3 3 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ18 3 3 -1 1 1 0 0 3 -3 -3 -1 0 0 3 3 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ19 3 3 -1 -1 1 0 0 3 3 -3 -1 0 0 -3 -3 1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ20 4 -4 0 0 0 -2 -2 0 0 0 0 2 2 -2√2 2√2 0 0 0 0 0 0 0 -√2 √2 -√2 √2 orthogonal faithful ρ21 4 -4 0 0 0 -2 -2 0 0 0 0 2 2 2√2 -2√2 0 0 0 0 0 0 0 √2 -√2 √2 -√2 orthogonal faithful ρ22 4 -4 0 0 0 1-√-3 1+√-3 0 0 0 0 -1+√-3 -1-√-3 2√2 -2√2 0 0 0 0 0 0 0 -ζ83ζ32+ζ8ζ32 -ζ87ζ32+ζ85ζ32 -ζ83ζ3+ζ8ζ3 -ζ87ζ3+ζ85ζ3 complex faithful ρ23 4 -4 0 0 0 1-√-3 1+√-3 0 0 0 0 -1+√-3 -1-√-3 -2√2 2√2 0 0 0 0 0 0 0 -ζ87ζ32+ζ85ζ32 -ζ83ζ32+ζ8ζ32 -ζ87ζ3+ζ85ζ3 -ζ83ζ3+ζ8ζ3 complex faithful ρ24 4 -4 0 0 0 1+√-3 1-√-3 0 0 0 0 -1-√-3 -1+√-3 2√2 -2√2 0 0 0 0 0 0 0 -ζ83ζ3+ζ8ζ3 -ζ87ζ3+ζ85ζ3 -ζ83ζ32+ζ8ζ32 -ζ87ζ32+ζ85ζ32 complex faithful ρ25 4 -4 0 0 0 1+√-3 1-√-3 0 0 0 0 -1-√-3 -1+√-3 -2√2 2√2 0 0 0 0 0 0 0 -ζ87ζ3+ζ85ζ3 -ζ83ζ3+ζ8ζ3 -ζ87ζ32+ζ85ζ32 -ζ83ζ32+ζ8ζ32 complex faithful ρ26 6 6 2 0 0 0 0 -6 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4×A4

Smallest permutation representation of Q16.A4
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 39 5 35)(2 38 6 34)(3 37 7 33)(4 36 8 40)(9 46 13 42)(10 45 14 41)(11 44 15 48)(12 43 16 47)(17 28 21 32)(18 27 22 31)(19 26 23 30)(20 25 24 29)
(1 3 5 7)(2 4 6 8)(9 44 13 48)(10 45 14 41)(11 46 15 42)(12 47 16 43)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)(33 39 37 35)(34 40 38 36)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 46 13 42)(10 47 14 43)(11 48 15 44)(12 41 16 45)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)
(1 26 43)(2 27 44)(3 28 45)(4 29 46)(5 30 47)(6 31 48)(7 32 41)(8 25 42)(9 40 24)(10 33 17)(11 34 18)(12 35 19)(13 36 20)(14 37 21)(15 38 22)(16 39 23)```

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

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

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

Matrix representation of Q16.A4 in GL4(𝔽7) generated by

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

Q16.A4 in GAP, Magma, Sage, TeX

`Q_{16}.A_4`
`% in TeX`

`G:=Group("Q16.A4");`
`// GroupNames label`

`G:=SmallGroup(192,1017);`
`// by ID`

`G=gap.SmallGroup(192,1017);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,672,197,680,3027,1522,248,438,172,775,285,124]);`
`// Polycyclic`

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

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