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## G = Q8.14D6order 96 = 25·3

### 4th non-split extension by Q8 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8.14D6
 Chief series C1 — C3 — C6 — C12 — Dic6 — C2×Dic6 — Q8.14D6
 Lower central C3 — C6 — C12 — Q8.14D6
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for Q8.14D6
G = < a,b,c,d | a4=c6=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 122 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, Q8.14D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, C2×C3⋊D4, Q8.14D6

Character table of Q8.14D6

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E size 1 1 2 4 2 2 2 4 12 12 2 4 4 4 12 12 2 2 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 -2 -1 2 2 -2 0 0 -1 -1 1 1 0 0 -1 -1 1 1 -1 orthogonal lifted from D6 ρ10 2 2 -2 2 -1 -2 2 -2 0 0 -1 1 -1 -1 0 0 1 1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 -1 2 2 2 0 0 -1 -1 -1 -1 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 0 2 -2 -2 0 0 0 2 2 0 0 0 0 -2 -2 0 0 -2 orthogonal lifted from D4 ρ13 2 2 -2 0 2 2 -2 0 0 0 2 -2 0 0 0 0 2 2 0 0 -2 orthogonal lifted from D4 ρ14 2 2 -2 -2 -1 -2 2 2 0 0 -1 1 1 1 0 0 1 1 -1 -1 -1 orthogonal lifted from D6 ρ15 2 2 -2 0 -1 2 -2 0 0 0 -1 1 -√-3 √-3 0 0 -1 -1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ16 2 2 2 0 -1 -2 -2 0 0 0 -1 -1 -√-3 √-3 0 0 1 1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ17 2 2 -2 0 -1 2 -2 0 0 0 -1 1 √-3 -√-3 0 0 -1 -1 -√-3 √-3 1 complex lifted from C3⋊D4 ρ18 2 2 2 0 -1 -2 -2 0 0 0 -1 -1 √-3 -√-3 0 0 1 1 √-3 -√-3 1 complex lifted from C3⋊D4 ρ19 4 -4 0 0 4 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ20 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 -2√3 2√3 0 0 0 symplectic faithful, Schur index 2 ρ21 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 0 0 2√3 -2√3 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of Q8.14D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 10 0 0 0 0 29 72 0 0 0 0 32 14 1 71 0 0 16 14 1 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 52 51 3 0 0 0 0 46 14 59 0 0 23 13 2 19 0 0 23 13 48 46
,
 8 0 0 0 0 0 71 64 0 0 0 0 0 0 52 51 3 0 0 0 48 46 0 14 0 0 35 13 21 54 0 0 29 13 21 27
,
 72 28 0 0 0 0 0 1 0 0 0 0 0 0 40 14 0 36 0 0 69 41 11 0 0 0 56 38 32 33 0 0 24 5 16 33

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,29,32,16,0,0,10,72,14,14,0,0,0,0,1,1,0,0,0,0,71,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,52,0,23,23,0,0,51,46,13,13,0,0,3,14,2,48,0,0,0,59,19,46],[8,71,0,0,0,0,0,64,0,0,0,0,0,0,52,48,35,29,0,0,51,46,13,13,0,0,3,0,21,21,0,0,0,14,54,27],[72,0,0,0,0,0,28,1,0,0,0,0,0,0,40,69,56,24,0,0,14,41,38,5,0,0,0,11,32,16,0,0,36,0,33,33] >;`

Q8.14D6 in GAP, Magma, Sage, TeX

`Q_8._{14}D_6`
`% in TeX`

`G:=Group("Q8.14D6");`
`// GroupNames label`

`G:=SmallGroup(96,158);`
`// by ID`

`G=gap.SmallGroup(96,158);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,188,579,159,69,2309]);`
`// Polycyclic`

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

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