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## G = Q8.8D12order 192 = 26·3

### 3rd non-split extension by Q8 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Q8.8D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C2×C24⋊C2 — Q8.8D12
 Lower central C3 — C6 — C2×C12 — Q8.8D12
 Upper central C1 — C2 — C2×C4 — C8○D4

Generators and relations for Q8.8D12
G = < a,b,c,d | a4=1, b2=c12=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c11 >

Subgroups: 320 in 104 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C24, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C24⋊C2, C4.Dic3, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, C2×D12, C3×C4○D4, D4.3D4, C24.C4, C12.46D4, C12.47D4, C2×C24⋊C2, D4⋊D6, Q8.14D6, C3×C8○D4, Q8.8D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, D4.3D4, C127D4, Q8.8D12

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 12D 12E 24A 24B 24C 24D 24E ··· 24J order 1 2 2 2 2 3 4 4 4 4 6 6 6 6 8 8 8 8 8 8 8 12 12 12 12 12 24 24 24 24 24 ··· 24 size 1 1 2 4 24 2 2 2 4 24 2 4 4 4 2 2 4 4 4 24 24 2 2 4 4 4 2 2 2 2 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 D12 D12 C4○D12 D4.3D4 Q8.8D12 kernel Q8.8D12 C24.C4 C12.46D4 C12.47D4 C2×C24⋊C2 D4⋊D6 Q8.14D6 C3×C8○D4 C8○D4 C24 C3×D4 C3×Q8 C2×C8 M4(2) C4○D4 C2×C6 C8 D4 Q8 C22 C3 C1 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 4 2 2 4 2 4

Matrix representation of Q8.8D12 in GL4(𝔽73) generated by

 66 59 0 0 14 7 0 0 46 0 7 14 0 46 59 66
,
 27 0 59 45 0 27 28 14 0 0 46 0 0 0 0 46
,
 48 37 0 0 36 11 0 0 0 0 48 37 0 0 36 11
,
 48 37 0 0 62 25 0 0 16 16 36 25 0 57 62 37
`G:=sub<GL(4,GF(73))| [66,14,46,0,59,7,0,46,0,0,7,59,0,0,14,66],[27,0,0,0,0,27,0,0,59,28,46,0,45,14,0,46],[48,36,0,0,37,11,0,0,0,0,48,36,0,0,37,11],[48,62,16,0,37,25,16,57,0,0,36,62,0,0,25,37] >;`

Q8.8D12 in GAP, Magma, Sage, TeX

`Q_8._8D_{12}`
`% in TeX`

`G:=Group("Q8.8D12");`
`// GroupNames label`

`G:=SmallGroup(192,700);`
`// by ID`

`G=gap.SmallGroup(192,700);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,1123,297,136,1684,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=1,b^2=c^12=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^11>;`
`// generators/relations`

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