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## G = C16.D6order 192 = 26·3

### 1st non-split extension by C16 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C16.D6
 Chief series C1 — C3 — C6 — C12 — C24 — D24 — C4○D24 — C16.D6
 Lower central C3 — C6 — C12 — C24 — C16.D6
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for C16.D6
G = < a,b,c | a16=1, b6=c2=a8, bab-1=a9, cac-1=a-1, cbc-1=b5 >

Subgroups: 296 in 82 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C16, C2×C8, D8, SD16, Q16, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, M5(2), SD32, Q32, C2×Q16, C4○D8, C48, C24⋊C2, D24, Dic12, Dic12, Dic12, C2×C24, C2×Dic6, C4○D12, Q32⋊C2, C48⋊C2, Dic24, C3×M5(2), C4○D24, C2×Dic12, C16.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, C2×D8, D24, C2×D12, Q32⋊C2, C2×D24, C16.D6

Smallest permutation representation of C16.D6
On 96 points
Generators in S96
```(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 37 61 93 79 26 9 45 53 85 71 18)(2 46 62 86 80 19 10 38 54 94 72 27)(3 39 63 95 65 28 11 47 55 87 73 20)(4 48 64 88 66 21 12 40 56 96 74 29)(5 41 49 81 67 30 13 33 57 89 75 22)(6 34 50 90 68 23 14 42 58 82 76 31)(7 43 51 83 69 32 15 35 59 91 77 24)(8 36 52 92 70 25 16 44 60 84 78 17)
(1 26 9 18)(2 25 10 17)(3 24 11 32)(4 23 12 31)(5 22 13 30)(6 21 14 29)(7 20 15 28)(8 19 16 27)(33 75 41 67)(34 74 42 66)(35 73 43 65)(36 72 44 80)(37 71 45 79)(38 70 46 78)(39 69 47 77)(40 68 48 76)(49 89 57 81)(50 88 58 96)(51 87 59 95)(52 86 60 94)(53 85 61 93)(54 84 62 92)(55 83 63 91)(56 82 64 90)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 12A 12B 12C 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 2 3 4 4 4 4 4 6 6 8 8 8 12 12 12 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 24 2 2 2 24 24 24 2 4 2 2 4 2 2 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 D8 D12 D12 D24 D24 Q32⋊C2 C16.D6 kernel C16.D6 C48⋊C2 Dic24 C3×M5(2) C4○D24 C2×Dic12 M5(2) C24 C2×C12 C16 C2×C8 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 1 1 1 2 1 2 2 2 2 4 4 2 4

Matrix representation of C16.D6 in GL4(𝔽97) generated by

 87 79 95 0 71 62 0 95 89 8 10 18 12 30 26 35
,
 65 52 45 11 11 11 86 34 13 43 80 39 31 82 32 38
,
 65 52 45 11 21 34 63 52 6 83 57 57 74 39 32 38
`G:=sub<GL(4,GF(97))| [87,71,89,12,79,62,8,30,95,0,10,26,0,95,18,35],[65,11,13,31,52,11,43,82,45,86,80,32,11,34,39,38],[65,21,6,74,52,34,83,39,45,63,57,32,11,52,57,38] >;`

C16.D6 in GAP, Magma, Sage, TeX

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

`G:=Group("C16.D6");`
`// GroupNames label`

`G:=SmallGroup(192,468);`
`// by ID`

`G=gap.SmallGroup(192,468);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,387,142,1571,80,1684,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^16=1,b^6=c^2=a^8,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

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