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## G = Q8.10D12order 192 = 26·3

### 5th non-split extension by Q8 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Q8.10D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C2×Dic12 — Q8.10D12
 Lower central C3 — C6 — C2×C12 — Q8.10D12
 Upper central C1 — C2 — C2×C4 — C8○D4

Generators and relations for Q8.10D12
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=a2c11 >

Subgroups: 256 in 100 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×2], C8 [×3], C2×C4, C2×C4 [×3], D4, D4, Q8, Q8 [×4], Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2) [×3], SD16 [×2], Q16 [×4], C2×Q8 [×2], C4○D4, C3⋊C8 [×2], C24 [×2], C24, Dic6 [×4], C2×Dic3 [×2], C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4.10D4 [×2], C8.C4, C8○D4, C2×Q16, C8.C22 [×2], Dic12 [×2], C4.Dic3 [×2], D4.S3 [×2], C3⋊Q16 [×2], C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6 [×2], C3×C4○D4, D4.5D4, C24.C4, C12.47D4 [×2], C2×Dic12, Q8.14D6 [×2], C3×C8○D4, Q8.10D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, D4.5D4, C127D4, Q8.10D12

Smallest permutation representation of Q8.10D12
On 96 points
Generators in S96
```(1 66 13 54)(2 67 14 55)(3 68 15 56)(4 69 16 57)(5 70 17 58)(6 71 18 59)(7 72 19 60)(8 49 20 61)(9 50 21 62)(10 51 22 63)(11 52 23 64)(12 53 24 65)(25 95 37 83)(26 96 38 84)(27 73 39 85)(28 74 40 86)(29 75 41 87)(30 76 42 88)(31 77 43 89)(32 78 44 90)(33 79 45 91)(34 80 46 92)(35 81 47 93)(36 82 48 94)
(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 77 37 89)(26 78 38 90)(27 79 39 91)(28 80 40 92)(29 81 41 93)(30 82 42 94)(31 83 43 95)(32 84 44 96)(33 85 45 73)(34 86 46 74)(35 87 47 75)(36 88 48 76)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 92 13 80)(2 91 14 79)(3 90 15 78)(4 89 16 77)(5 88 17 76)(6 87 18 75)(7 86 19 74)(8 85 20 73)(9 84 21 96)(10 83 22 95)(11 82 23 94)(12 81 24 93)(25 51 37 63)(26 50 38 62)(27 49 39 61)(28 72 40 60)(29 71 41 59)(30 70 42 58)(31 69 43 57)(32 68 44 56)(33 67 45 55)(34 66 46 54)(35 65 47 53)(36 64 48 52)```

`G:=sub<Sym(96)| (1,66,13,54)(2,67,14,55)(3,68,15,56)(4,69,16,57)(5,70,17,58)(6,71,18,59)(7,72,19,60)(8,49,20,61)(9,50,21,62)(10,51,22,63)(11,52,23,64)(12,53,24,65)(25,95,37,83)(26,96,38,84)(27,73,39,85)(28,74,40,86)(29,75,41,87)(30,76,42,88)(31,77,43,89)(32,78,44,90)(33,79,45,91)(34,80,46,92)(35,81,47,93)(36,82,48,94), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,77,37,89)(26,78,38,90)(27,79,39,91)(28,80,40,92)(29,81,41,93)(30,82,42,94)(31,83,43,95)(32,84,44,96)(33,85,45,73)(34,86,46,74)(35,87,47,75)(36,88,48,76)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,92,13,80)(2,91,14,79)(3,90,15,78)(4,89,16,77)(5,88,17,76)(6,87,18,75)(7,86,19,74)(8,85,20,73)(9,84,21,96)(10,83,22,95)(11,82,23,94)(12,81,24,93)(25,51,37,63)(26,50,38,62)(27,49,39,61)(28,72,40,60)(29,71,41,59)(30,70,42,58)(31,69,43,57)(32,68,44,56)(33,67,45,55)(34,66,46,54)(35,65,47,53)(36,64,48,52)>;`

`G:=Group( (1,66,13,54)(2,67,14,55)(3,68,15,56)(4,69,16,57)(5,70,17,58)(6,71,18,59)(7,72,19,60)(8,49,20,61)(9,50,21,62)(10,51,22,63)(11,52,23,64)(12,53,24,65)(25,95,37,83)(26,96,38,84)(27,73,39,85)(28,74,40,86)(29,75,41,87)(30,76,42,88)(31,77,43,89)(32,78,44,90)(33,79,45,91)(34,80,46,92)(35,81,47,93)(36,82,48,94), (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,77,37,89)(26,78,38,90)(27,79,39,91)(28,80,40,92)(29,81,41,93)(30,82,42,94)(31,83,43,95)(32,84,44,96)(33,85,45,73)(34,86,46,74)(35,87,47,75)(36,88,48,76)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,92,13,80)(2,91,14,79)(3,90,15,78)(4,89,16,77)(5,88,17,76)(6,87,18,75)(7,86,19,74)(8,85,20,73)(9,84,21,96)(10,83,22,95)(11,82,23,94)(12,81,24,93)(25,51,37,63)(26,50,38,62)(27,49,39,61)(28,72,40,60)(29,71,41,59)(30,70,42,58)(31,69,43,57)(32,68,44,56)(33,67,45,55)(34,66,46,54)(35,65,47,53)(36,64,48,52) );`

`G=PermutationGroup([(1,66,13,54),(2,67,14,55),(3,68,15,56),(4,69,16,57),(5,70,17,58),(6,71,18,59),(7,72,19,60),(8,49,20,61),(9,50,21,62),(10,51,22,63),(11,52,23,64),(12,53,24,65),(25,95,37,83),(26,96,38,84),(27,73,39,85),(28,74,40,86),(29,75,41,87),(30,76,42,88),(31,77,43,89),(32,78,44,90),(33,79,45,91),(34,80,46,92),(35,81,47,93),(36,82,48,94)], [(1,7,13,19),(2,8,14,20),(3,9,15,21),(4,10,16,22),(5,11,17,23),(6,12,18,24),(25,77,37,89),(26,78,38,90),(27,79,39,91),(28,80,40,92),(29,81,41,93),(30,82,42,94),(31,83,43,95),(32,84,44,96),(33,85,45,73),(34,86,46,74),(35,87,47,75),(36,88,48,76),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,92,13,80),(2,91,14,79),(3,90,15,78),(4,89,16,77),(5,88,17,76),(6,87,18,75),(7,86,19,74),(8,85,20,73),(9,84,21,96),(10,83,22,95),(11,82,23,94),(12,81,24,93),(25,51,37,63),(26,50,38,62),(27,49,39,61),(28,72,40,60),(29,71,41,59),(30,70,42,58),(31,69,43,57),(32,68,44,56),(33,67,45,55),(34,66,46,54),(35,65,47,53),(36,64,48,52)])`

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 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 3 4 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 2 2 2 4 24 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 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 D4.5D4 Q8.10D12 kernel Q8.10D12 C24.C4 C12.47D4 C2×Dic12 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 2 1 2 1 1 2 1 1 1 1 1 2 4 2 2 4 2 4

Matrix representation of Q8.10D12 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 72 71 0 0 72 0 0 0 0 0 1 1 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 72 72 1 2 0 0 0 1 72 72
,
 64 0 0 0 0 0 31 8 0 0 0 0 0 0 57 16 0 0 0 0 57 57 0 0 0 0 16 16 41 41 0 0 57 0 16 0
,
 14 43 0 0 0 0 43 59 0 0 0 0 0 0 7 0 66 6 0 0 13 0 60 53 0 0 60 67 66 6 0 0 13 66 0 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,72,1,0,0,0,1,0,1,0,0,1,72,0,0,0,0,0,71,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,1,0,72,1,0,0,0,0,1,72,0,0,0,0,2,72],[64,31,0,0,0,0,0,8,0,0,0,0,0,0,57,57,16,57,0,0,16,57,16,0,0,0,0,0,41,16,0,0,0,0,41,0],[14,43,0,0,0,0,43,59,0,0,0,0,0,0,7,13,60,13,0,0,0,0,67,66,0,0,66,60,66,0,0,0,6,53,6,0] >;`

Q8.10D12 in GAP, Magma, Sage, TeX

`Q_8._{10}D_{12}`
`% in TeX`

`G:=Group("Q8.10D12");`
`// GroupNames label`

`G:=SmallGroup(192,702);`
`// by ID`

`G=gap.SmallGroup(192,702);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,344,254,1123,297,136,1684,102,6278]);`
`// 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=a^2*c^11>;`
`// generators/relations`

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