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## G = Q8.11D6order 96 = 25·3

### 1st non-split extension by Q8 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8.11D6
 Chief series C1 — C3 — C6 — C12 — D12 — C4○D12 — Q8.11D6
 Lower central C3 — C6 — C12 — Q8.11D6
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for Q8.11D6
G = < a,b,c,d | a4=1, b2=c6=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab, dcd-1=c5 >

Subgroups: 130 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C8.C22, C4.Dic3, Q82S3, C3⋊Q16, C4○D12, C6×Q8, Q8.11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, C2×C3⋊D4, Q8.11D6

Character table of Q8.11D6

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F size 1 1 2 12 2 2 2 4 4 12 2 2 2 12 12 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 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 0 -1 2 2 2 2 0 -1 -1 -1 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 0 -1 -2 2 2 -2 0 1 -1 1 0 0 1 1 -1 -1 -1 1 orthogonal lifted from D6 ρ11 2 2 2 0 2 -2 -2 0 0 0 2 2 2 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ12 2 2 -2 0 2 2 -2 0 0 0 -2 2 -2 0 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ13 2 2 -2 0 -1 -2 2 -2 2 0 1 -1 1 0 0 -1 -1 1 1 -1 1 orthogonal lifted from D6 ρ14 2 2 2 0 -1 2 2 -2 -2 0 -1 -1 -1 0 0 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ15 2 2 2 0 -1 -2 -2 0 0 0 -1 -1 -1 0 0 √-3 -√-3 -√-3 √-3 1 1 complex lifted from C3⋊D4 ρ16 2 2 -2 0 -1 2 -2 0 0 0 1 -1 1 0 0 √-3 -√-3 √-3 -√-3 1 -1 complex lifted from C3⋊D4 ρ17 2 2 -2 0 -1 2 -2 0 0 0 1 -1 1 0 0 -√-3 √-3 -√-3 √-3 1 -1 complex lifted from C3⋊D4 ρ18 2 2 2 0 -1 -2 -2 0 0 0 -1 -1 -1 0 0 -√-3 √-3 √-3 -√-3 1 1 complex lifted from C3⋊D4 ρ19 4 -4 0 0 4 0 0 0 0 0 0 -4 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√-3 2 2√-3 0 0 0 0 0 0 0 0 complex faithful ρ21 4 -4 0 0 -2 0 0 0 0 0 2√-3 2 -2√-3 0 0 0 0 0 0 0 0 complex faithful

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

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

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

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

Matrix representation of Q8.11D6 in GL4(𝔽7) generated by

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

Q8.11D6 in GAP, Magma, Sage, TeX

`Q_8._{11}D_6`
`% in TeX`

`G:=Group("Q8.11D6");`
`// GroupNames label`

`G:=SmallGroup(96,149);`
`// by ID`

`G=gap.SmallGroup(96,149);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,86,579,159,69,2309]);`
`// 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=a^2*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;`
`// generators/relations`

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