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## G = C8.8D12order 192 = 26·3

### 4th non-split extension by C8 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C8.8D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C42⋊7S3 — C8.8D12
 Lower central C3 — C6 — C2×C12 — C8.8D12
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C8.8D12
G = < a,b,c | a8=b12=1, c2=a4, ab=ba, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 472 in 130 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C42, C22⋊C4 [×4], C2×C8 [×2], D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], C24 [×4], Dic6 [×4], D12 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C24⋊C2 [×4], D24 [×2], Dic12 [×2], D6⋊C4 [×4], C4×C12, C2×C24 [×2], C2×Dic6 [×2], C2×D12 [×2], C8.12D4, C4×C24, C427S3 [×2], C2×C24⋊C2 [×2], C2×D24, C2×Dic12, C8.8D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], D12 [×6], C22×S3, C41D4, C4○D8 [×2], C2×D12 [×3], C8.12D4, C4⋊D12, C4○D24 [×2], C8.8D12

Smallest permutation representation of C8.8D12
On 96 points
Generators in S96
```(1 29 47 67 73 57 20 94)(2 30 48 68 74 58 21 95)(3 31 37 69 75 59 22 96)(4 32 38 70 76 60 23 85)(5 33 39 71 77 49 24 86)(6 34 40 72 78 50 13 87)(7 35 41 61 79 51 14 88)(8 36 42 62 80 52 15 89)(9 25 43 63 81 53 16 90)(10 26 44 64 82 54 17 91)(11 27 45 65 83 55 18 92)(12 28 46 66 84 56 19 93)
(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 84 73 12)(2 11 74 83)(3 82 75 10)(4 9 76 81)(5 80 77 8)(6 7 78 79)(13 41 40 14)(15 39 42 24)(16 23 43 38)(17 37 44 22)(18 21 45 48)(19 47 46 20)(25 70 53 85)(26 96 54 69)(27 68 55 95)(28 94 56 67)(29 66 57 93)(30 92 58 65)(31 64 59 91)(32 90 60 63)(33 62 49 89)(34 88 50 61)(35 72 51 87)(36 86 52 71)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D12 D12 C4○D8 C4○D24 kernel C8.8D12 C4×C24 C42⋊7S3 C2×C24⋊C2 C2×D24 C2×Dic12 C4×C8 C24 C2×C12 C42 C2×C8 C8 C2×C4 C6 C2 # reps 1 1 2 2 1 1 1 4 2 1 2 8 4 8 16

Matrix representation of C8.8D12 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 0 48 0 0 38 32
,
 7 66 0 0 7 14 0 0 0 0 46 8 0 0 55 27
,
 7 66 0 0 59 66 0 0 0 0 46 8 0 0 0 27
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,38,0,0,48,32],[7,7,0,0,66,14,0,0,0,0,46,55,0,0,8,27],[7,59,0,0,66,66,0,0,0,0,46,0,0,0,8,27] >;`

C8.8D12 in GAP, Magma, Sage, TeX

`C_8._8D_{12}`
`% in TeX`

`G:=Group("C8.8D12");`
`// GroupNames label`

`G:=SmallGroup(192,255);`
`// by ID`

`G=gap.SmallGroup(192,255);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,58,1123,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^12=1,c^2=a^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;`
`// generators/relations`

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