Copied to
clipboard

## G = C16.12D6order 192 = 26·3

### 9th non-split extension by C16 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C16.12D6
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — C8○D12 — C16.12D6
 Lower central C3 — C6 — C16.12D6
 Upper central C1 — C8 — M5(2)

Generators and relations for C16.12D6
G = < a,b,c | a16=b6=1, c2=a8, bab-1=a9, ac=ca, cbc-1=a8b-1 >

Subgroups: 152 in 84 conjugacy classes, 51 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C16, C16, C2×C8, C2×C8, M4(2), C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C16, M5(2), M5(2), C8○D4, C3⋊C16, C48, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, D4○C16, S3×C16, D6.C8, C2×C3⋊C16, C3×M5(2), C8○D12, C16.12D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C4×S3, C22×S3, C22×C8, S3×C8, S3×C2×C4, D4○C16, S3×C2×C8, C16.12D6

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 16A ··· 16H 16I ··· 16P 16Q 16R 16S 16T 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 2 2 3 4 4 4 4 4 6 6 8 8 8 8 8 8 8 8 8 8 12 12 12 16 ··· 16 16 ··· 16 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 6 6 2 1 1 2 6 6 2 4 1 1 1 1 2 2 6 6 6 6 2 2 4 2 ··· 2 3 ··· 3 6 6 6 6 2 2 2 2 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 S3 D6 D6 C4×S3 C4×S3 S3×C8 S3×C8 D4○C16 C16.12D6 kernel C16.12D6 S3×C16 D6.C8 C2×C3⋊C16 C3×M5(2) C8○D12 C8⋊S3 C4.Dic3 C4○D12 Dic6 D12 C3⋊D4 M5(2) C16 C2×C8 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 4 2 2 4 4 8 1 2 1 2 2 4 4 8 4

Matrix representation of C16.12D6 in GL4(𝔽97) generated by

 47 0 0 0 0 47 0 0 0 0 12 0 0 0 0 85
,
 0 96 0 0 1 96 0 0 0 0 0 89 0 0 12 0
,
 1 0 0 0 1 96 0 0 0 0 22 0 0 0 0 75
`G:=sub<GL(4,GF(97))| [47,0,0,0,0,47,0,0,0,0,12,0,0,0,0,85],[0,1,0,0,96,96,0,0,0,0,0,12,0,0,89,0],[1,1,0,0,0,96,0,0,0,0,22,0,0,0,0,75] >;`

C16.12D6 in GAP, Magma, Sage, TeX

`C_{16}._{12}D_6`
`% in TeX`

`G:=Group("C16.12D6");`
`// GroupNames label`

`G:=SmallGroup(192,466);`
`// by ID`

`G=gap.SmallGroup(192,466);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,58,80,102,6278]);`
`// Polycyclic`

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

׿
×
𝔽