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## G = Q16.D6order 192 = 26·3

### 3rd non-split extension by Q16 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — Q16.D6
 Chief series C1 — C3 — C6 — C12 — C24 — D24 — C4○D24 — Q16.D6
 Lower central C3 — C6 — C12 — C24 — Q16.D6
 Upper central C1 — C4 — C2×C4 — C2×C8 — C4○D8

Generators and relations for Q16.D6
G = < a,b,c,d | a8=1, b2=c6=d2=a4, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c5 >

Subgroups: 264 in 84 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C2×C6, C16, C2×C8, D8, D8, SD16, Q16, Q16, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C2×C16, D16, SD32, Q32, C4○D8, C4○D8, C3⋊C16, C24⋊C2, D24, Dic12, C2×C24, C3×D8, C3×SD16, C3×Q16, C4○D12, C3×C4○D4, C4○D16, C2×C3⋊C16, C3⋊D16, D8.S3, C8.6D6, C3⋊Q32, C4○D24, C3×C4○D8, Q16.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, C4○D16, C2×D4⋊S3, Q16.D6

Smallest permutation representation of Q16.D6
On 96 points
Generators in S96
```(1 50 16 35 7 56 22 29)(2 51 17 36 8 57 23 30)(3 52 18 25 9 58 24 31)(4 53 19 26 10 59 13 32)(5 54 20 27 11 60 14 33)(6 55 21 28 12 49 15 34)(37 72 81 90 43 66 75 96)(38 61 82 91 44 67 76 85)(39 62 83 92 45 68 77 86)(40 63 84 93 46 69 78 87)(41 64 73 94 47 70 79 88)(42 65 74 95 48 71 80 89)
(1 79 7 73)(2 80 8 74)(3 81 9 75)(4 82 10 76)(5 83 11 77)(6 84 12 78)(13 44 19 38)(14 45 20 39)(15 46 21 40)(16 47 22 41)(17 48 23 42)(18 37 24 43)(25 96 31 90)(26 85 32 91)(27 86 33 92)(28 87 34 93)(29 88 35 94)(30 89 36 95)(49 69 55 63)(50 70 56 64)(51 71 57 65)(52 72 58 66)(53 61 59 67)(54 62 60 68)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 24 19 18)(14 17 20 23)(15 22 21 16)(25 59 31 53)(26 52 32 58)(27 57 33 51)(28 50 34 56)(29 55 35 49)(30 60 36 54)(37 67 43 61)(38 72 44 66)(39 65 45 71)(40 70 46 64)(41 63 47 69)(42 68 48 62)(73 87 79 93)(74 92 80 86)(75 85 81 91)(76 90 82 96)(77 95 83 89)(78 88 84 94)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D 12E 16A ··· 16H 24A 24B 24C 24D order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 6 8 8 8 8 12 12 12 12 12 16 ··· 16 24 24 24 24 size 1 1 2 8 24 2 1 1 2 8 24 2 4 8 8 2 2 2 2 2 2 4 8 8 6 ··· 6 4 4 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 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D8 D8 C3⋊D4 C3⋊D4 C4○D16 D4⋊S3 D4⋊S3 Q16.D6 kernel Q16.D6 C2×C3⋊C16 C3⋊D16 D8.S3 C8.6D6 C3⋊Q32 C4○D24 C3×C4○D8 C4○D8 C24 C2×C12 C2×C8 D8 Q16 C12 C2×C6 C8 C2×C4 C3 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 8 1 1 4

Matrix representation of Q16.D6 in GL4(𝔽97) generated by

 7 90 0 0 7 7 0 0 0 0 96 0 0 0 0 96
,
 53 87 0 0 87 44 0 0 0 0 41 15 0 0 82 56
,
 22 0 0 0 0 22 0 0 0 0 0 1 0 0 96 1
,
 22 0 0 0 0 75 0 0 0 0 96 1 0 0 0 1
`G:=sub<GL(4,GF(97))| [7,7,0,0,90,7,0,0,0,0,96,0,0,0,0,96],[53,87,0,0,87,44,0,0,0,0,41,82,0,0,15,56],[22,0,0,0,0,22,0,0,0,0,0,96,0,0,1,1],[22,0,0,0,0,75,0,0,0,0,96,0,0,0,1,1] >;`

Q16.D6 in GAP, Magma, Sage, TeX

`Q_{16}.D_6`
`% in TeX`

`G:=Group("Q16.D6");`
`// GroupNames label`

`G:=SmallGroup(192,753);`
`// by ID`

`G=gap.SmallGroup(192,753);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,675,185,192,1684,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=1,b^2=c^6=d^2=a^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^5>;`
`// generators/relations`

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