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## G = Q8.15D6order 96 = 25·3

### 1st non-split extension by Q8 of D6 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Q8.15D6
 Chief series C1 — C3 — C6 — D6 — C4×S3 — S3×Q8 — Q8.15D6
 Lower central C3 — C6 — Q8.15D6
 Upper central C1 — C2 — C2×Q8

Generators and relations for Q8.15D6
G = < a,b,c,d | a4=1, b2=c6=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c5 >

Subgroups: 274 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, Dic3, C12, D6, C2×C6, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, 2- 1+4, C4○D12, S3×Q8, Q83S3, C6×Q8, Q8.15D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S3×C23, Q8.15D6

Character table of Q8.15D6

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 2 6 6 6 6 2 2 2 2 2 2 2 6 6 6 6 2 2 2 4 4 4 4 4 4 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 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 -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 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 -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 -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 -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 -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 -1 1 1 -1 linear of order 2 ρ9 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 1 linear of order 2 ρ10 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 -1 linear of order 2 ρ11 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 1 linear of order 2 ρ12 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 -1 linear of order 2 ρ13 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 1 linear of order 2 ρ14 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 -1 linear of order 2 ρ15 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 1 linear of order 2 ρ16 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 -1 linear of order 2 ρ17 2 2 -2 0 0 0 0 -1 2 2 -2 2 -2 -2 0 0 0 0 1 1 -1 -1 1 1 -1 -1 1 orthogonal lifted from D6 ρ18 2 2 2 0 0 0 0 -1 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 2 0 0 0 0 -1 2 -2 -2 -2 -2 2 0 0 0 0 -1 -1 -1 1 -1 1 -1 1 1 orthogonal lifted from D6 ρ20 2 2 -2 0 0 0 0 -1 2 -2 2 -2 2 -2 0 0 0 0 1 1 -1 1 1 -1 -1 1 -1 orthogonal lifted from D6 ρ21 2 2 -2 0 0 0 0 -1 -2 -2 2 2 -2 2 0 0 0 0 1 1 -1 1 -1 1 1 -1 -1 orthogonal lifted from D6 ρ22 2 2 2 0 0 0 0 -1 -2 -2 -2 2 2 -2 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ23 2 2 2 0 0 0 0 -1 -2 2 2 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 1 1 1 1 -1 orthogonal lifted from D6 ρ24 2 2 -2 0 0 0 0 -1 -2 2 -2 -2 2 2 0 0 0 0 1 1 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ25 4 -4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 -4 0 0 0 0 0 0 symplectic lifted from 2- 1+4, Schur index 2 ρ26 4 -4 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 -2√-3 2√-3 2 0 0 0 0 0 0 complex faithful ρ27 4 -4 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2√-3 -2√-3 2 0 0 0 0 0 0 complex faithful

Smallest permutation representation of Q8.15D6
On 48 points
Generators in S48
```(1 42 7 48)(2 43 8 37)(3 44 9 38)(4 45 10 39)(5 46 11 40)(6 47 12 41)(13 25 19 31)(14 26 20 32)(15 27 21 33)(16 28 22 34)(17 29 23 35)(18 30 24 36)
(1 26 7 32)(2 33 8 27)(3 28 9 34)(4 35 10 29)(5 30 11 36)(6 25 12 31)(13 41 19 47)(14 48 20 42)(15 43 21 37)(16 38 22 44)(17 45 23 39)(18 40 24 46)
(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 19 7 13)(2 24 8 18)(3 17 9 23)(4 22 10 16)(5 15 11 21)(6 20 12 14)(25 48 31 42)(26 41 32 47)(27 46 33 40)(28 39 34 45)(29 44 35 38)(30 37 36 43)```

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

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

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

Matrix representation of Q8.15D6 in GL4(𝔽7) generated by

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

Q8.15D6 in GAP, Magma, Sage, TeX

`Q_8._{15}D_6`
`% in TeX`

`G:=Group("Q8.15D6");`
`// GroupNames label`

`G:=SmallGroup(96,214);`
`// by ID`

`G=gap.SmallGroup(96,214);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,188,86,579,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=1,b^2=c^6=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^5>;`
`// generators/relations`

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