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## G = Q8.9D12order 192 = 26·3

### 4th non-split extension by Q8 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Q8.9D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C2×D24 — Q8.9D12
 Lower central C3 — C6 — C2×C12 — Q8.9D12
 Upper central C1 — C2 — C2×C4 — C8○D4

Generators and relations for Q8.9D12
G = < a,b,c,d | a4=d2=1, b2=c12=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c11 >

Subgroups: 384 in 108 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C3⋊C8, C24, C24, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, D24, C4.Dic3, D4⋊S3, Q82S3, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×D12, C3×C4○D4, D4.4D4, C24.C4, C12.46D4, C2×D24, D4⋊D6, C3×C8○D4, Q8.9D12
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, D4.4D4, C127D4, Q8.9D12

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 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 2 3 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 24 2 2 2 4 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 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 D12 D12 C4○D12 D4.4D4 Q8.9D12 kernel Q8.9D12 C24.C4 C12.46D4 C2×D24 D4⋊D6 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 2 1 2 1 1 2 1 1 1 1 1 2 4 2 2 4 2 4

Matrix representation of Q8.9D12 in GL4(𝔽73) generated by

 7 14 0 0 59 66 0 0 0 0 66 59 0 0 14 7
,
 0 0 66 59 0 0 14 7 66 59 0 0 14 7 0 0
,
 23 18 0 0 55 5 0 0 0 0 23 18 0 0 55 5
,
 23 18 0 0 68 50 0 0 0 0 55 50 0 0 68 18
`G:=sub<GL(4,GF(73))| [7,59,0,0,14,66,0,0,0,0,66,14,0,0,59,7],[0,0,66,14,0,0,59,7,66,14,0,0,59,7,0,0],[23,55,0,0,18,5,0,0,0,0,23,55,0,0,18,5],[23,68,0,0,18,50,0,0,0,0,55,68,0,0,50,18] >;`

Q8.9D12 in GAP, Magma, Sage, TeX

`Q_8._9D_{12}`
`% in TeX`

`G:=Group("Q8.9D12");`
`// GroupNames label`

`G:=SmallGroup(192,701);`
`// by ID`

`G=gap.SmallGroup(192,701);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,1123,297,136,1684,102,6278]);`
`// Polycyclic`

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

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