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## G = C12.16D8order 192 = 26·3

### 16th non-split extension by C12 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.16D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C4×C3⋊C8 — C12.16D8
 Lower central C3 — C6 — C2×C12 — C12.16D8
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C12.16D8
G = < a,b,c | a12=b8=1, c2=a6, bab-1=a5, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 336 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C4 [×2], C22, C22 [×6], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], C23 [×2], Dic3 [×2], C12 [×6], C2×C6, C2×C6 [×6], C42, C4⋊C4 [×3], C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C3×D4 [×8], C22×C6 [×2], C4×C8, D4⋊C4 [×4], C41D4, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C2×Dic6, C6×D4 [×2], C6×D4 [×2], C4.4D8, C4×C3⋊C8, D4⋊Dic3 [×4], C122Q8, C3×C41D4, C12.16D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C2×D8, C2×SD16, D4⋊S3 [×2], D4.S3 [×2], D42S3 [×2], C2×C3⋊D4, C4.4D8, C2×D4⋊S3, C2×D4.S3, C23.12D6, C12.16D8

Smallest permutation representation of C12.16D8
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 46 66 15 82 28 60 95)(2 39 67 20 83 33 49 88)(3 44 68 13 84 26 50 93)(4 37 69 18 73 31 51 86)(5 42 70 23 74 36 52 91)(6 47 71 16 75 29 53 96)(7 40 72 21 76 34 54 89)(8 45 61 14 77 27 55 94)(9 38 62 19 78 32 56 87)(10 43 63 24 79 25 57 92)(11 48 64 17 80 30 58 85)(12 41 65 22 81 35 59 90)
(1 86 7 92)(2 85 8 91)(3 96 9 90)(4 95 10 89)(5 94 11 88)(6 93 12 87)(13 81 19 75)(14 80 20 74)(15 79 21 73)(16 78 22 84)(17 77 23 83)(18 76 24 82)(25 66 31 72)(26 65 32 71)(27 64 33 70)(28 63 34 69)(29 62 35 68)(30 61 36 67)(37 54 43 60)(38 53 44 59)(39 52 45 58)(40 51 46 57)(41 50 47 56)(42 49 48 55)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - - image C1 C2 C2 C2 C2 S3 D4 D6 D6 D8 SD16 C4○D4 C3⋊D4 D4⋊S3 D4.S3 D4⋊2S3 kernel C12.16D8 C4×C3⋊C8 D4⋊Dic3 C12⋊2Q8 C3×C4⋊1D4 C4⋊1D4 C2×C12 C42 C2×D4 C12 C12 C12 C2×C4 C4 C4 C4 # reps 1 1 4 1 1 1 2 1 2 4 4 4 4 2 2 2

Matrix representation of C12.16D8 in GL6(𝔽73)

 1 2 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 8 0 0 0 0 0 0 64
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 16 16 0 0 0 0 57 16 0 0 0 0 0 0 0 2 0 0 0 0 36 0
,
 27 54 0 0 0 0 0 46 0 0 0 0 0 0 57 57 0 0 0 0 57 16 0 0 0 0 0 0 0 2 0 0 0 0 37 0

`G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,2,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,8,0,0,0,0,0,0,64],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,16,57,0,0,0,0,16,16,0,0,0,0,0,0,0,36,0,0,0,0,2,0],[27,0,0,0,0,0,54,46,0,0,0,0,0,0,57,57,0,0,0,0,57,16,0,0,0,0,0,0,0,37,0,0,0,0,2,0] >;`

C12.16D8 in GAP, Magma, Sage, TeX

`C_{12}._{16}D_8`
`% in TeX`

`G:=Group("C12.16D8");`
`// GroupNames label`

`G:=SmallGroup(192,629);`
`// by ID`

`G=gap.SmallGroup(192,629);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,590,135,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;`
`// generators/relations`

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