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## G = Q8○D12order 96 = 25·3

### Central product of Q8 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — Q8○D12
 Chief series C1 — C3 — C6 — D6 — C4×S3 — S3×Q8 — Q8○D12
 Lower central C3 — C6 — Q8○D12
 Upper central C1 — C2 — C4○D4

Generators and relations for Q8○D12
G = < a,b,c,d | a4=d2=1, b2=c6=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c5 >

Subgroups: 266 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C12, D6, C2×C6, C2×Q8, C4○D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, 2- 1+4, C2×Dic6, C4○D12, D42S3, S3×Q8, C3×C4○D4, Q8○D12
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2- 1+4, S3×C23, Q8○D12

Character table of Q8○D12

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 12A 12B 12C 12D 12E size 1 1 2 2 2 6 6 2 2 2 2 2 6 6 6 6 6 6 2 4 4 4 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 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 2 -2 0 0 -1 -2 -2 2 2 0 0 0 0 0 0 -1 1 -1 1 -1 -1 -1 1 1 orthogonal lifted from D6 ρ18 2 2 2 2 2 0 0 -1 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 -2 -2 2 0 0 -1 -2 2 2 -2 0 0 0 0 0 0 -1 1 1 -1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ20 2 2 2 -2 -2 0 0 -1 2 -2 2 -2 0 0 0 0 0 0 -1 -1 1 1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ21 2 2 -2 -2 -2 0 0 -1 2 2 -2 2 0 0 0 0 0 0 -1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from D6 ρ22 2 2 2 -2 2 0 0 -1 -2 -2 -2 2 0 0 0 0 0 0 -1 -1 1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ23 2 2 -2 2 2 0 0 -1 2 -2 -2 -2 0 0 0 0 0 0 -1 1 -1 -1 1 1 1 1 -1 orthogonal lifted from D6 ρ24 2 2 2 2 -2 0 0 -1 -2 2 -2 -2 0 0 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 -4 0 0 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 0 0 0 -2√3 2√3 0 0 0 symplectic faithful, Schur index 2 ρ27 4 -4 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2√3 -2√3 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of Q8○D12
On 48 points
Generators in S48
```(1 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 43 22 37)(17 44 23 38)(18 45 24 39)
(1 46 7 40)(2 47 8 41)(3 48 9 42)(4 37 10 43)(5 38 11 44)(6 39 12 45)(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 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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(25 27)(28 36)(29 35)(30 34)(31 33)(37 45)(38 44)(39 43)(40 42)(46 48)```

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

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

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

Matrix representation of Q8○D12 in GL4(𝔽13) generated by

 8 0 0 0 0 8 0 0 5 8 5 0 10 5 0 5
,
 8 0 1 1 0 8 11 1 0 0 5 0 0 0 0 5
,
 10 3 0 0 10 7 0 0 0 0 7 3 0 0 10 10
,
 1 1 0 0 0 12 0 0 0 0 12 1 0 0 0 1
`G:=sub<GL(4,GF(13))| [8,0,5,10,0,8,8,5,0,0,5,0,0,0,0,5],[8,0,0,0,0,8,0,0,1,11,5,0,1,1,0,5],[10,10,0,0,3,7,0,0,0,0,7,10,0,0,3,10],[1,0,0,0,1,12,0,0,0,0,12,0,0,0,1,1] >;`

Q8○D12 in GAP, Magma, Sage, TeX

`Q_8\circ D_{12}`
`% in TeX`

`G:=Group("Q8oD12");`
`// GroupNames label`

`G:=SmallGroup(96,217);`
`// by ID`

`G=gap.SmallGroup(96,217);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,188,86,579,69,2309]);`
`// Polycyclic`

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

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