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## G = C12.9D8order 192 = 26·3

### 9th non-split extension by C12 of D8 acting via D8/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.9D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×C12 — C12⋊C8 — C12.9D8
 Lower central C3 — C2×C6 — C2×C12 — C12.9D8
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C12.9D8
G = < a,b,c | a12=b8=1, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a9b-1 >

Subgroups: 240 in 84 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4, C22, C22 [×6], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C12 [×4], C12, C2×C6, C2×C6 [×6], C42, C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], C3⋊C8 [×2], C2×C12, C2×C12 [×2], C3×D4 [×6], C22×C6 [×2], C4⋊C8 [×2], C41D4, C2×C3⋊C8 [×2], C4×C12, C6×D4 [×2], C6×D4 [×2], C4.D8, C12⋊C8 [×2], C3×C41D4, C12.9D8
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D8 [×2], SD16 [×2], C2×Dic3, C3⋊D4 [×2], C4.D4, D4⋊C4 [×2], D4⋊S3 [×2], D4.S3 [×2], C6.D4, C4.D8, D4⋊Dic3 [×2], C12.D4, C12.9D8

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + - image C1 C2 C2 C4 S3 D4 D6 Dic3 D8 SD16 C3⋊D4 C4.D4 D4⋊S3 D4.S3 C12.D4 kernel C12.9D8 C12⋊C8 C3×C4⋊1D4 C6×D4 C4⋊1D4 C2×C12 C42 C2×D4 C12 C12 C2×C4 C6 C4 C4 C2 # reps 1 2 1 4 1 2 1 2 4 4 4 1 2 2 2

Matrix representation of C12.9D8 in GL6(𝔽73)

 0 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 65 0 0 0 0 0 25 9
,
 57 16 0 0 0 0 16 16 0 0 0 0 0 0 57 57 0 0 0 0 16 57 0 0 0 0 0 0 33 2 0 0 0 0 40 40
,
 57 16 0 0 0 0 57 57 0 0 0 0 0 0 57 57 0 0 0 0 57 16 0 0 0 0 0 0 33 2 0 0 0 0 39 40

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,65,25,0,0,0,0,0,9],[57,16,0,0,0,0,16,16,0,0,0,0,0,0,57,16,0,0,0,0,57,57,0,0,0,0,0,0,33,40,0,0,0,0,2,40],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,57,57,0,0,0,0,57,16,0,0,0,0,0,0,33,39,0,0,0,0,2,40] >;`

C12.9D8 in GAP, Magma, Sage, TeX

`C_{12}._9D_8`
`% in TeX`

`G:=Group("C12.9D8");`
`// GroupNames label`

`G:=SmallGroup(192,103);`
`// by ID`

`G=gap.SmallGroup(192,103);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,100,1571,570,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^8=1,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

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