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## G = C12.50D8order 192 = 26·3

### 4th non-split extension by C12 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12.50D8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C12⋊2Q8 — C12.50D8
 Lower central C3 — C6 — C2×C12 — C12.50D8
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for C12.50D8
G = < a,b,c | a12=b8=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 280 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×C6, D4⋊C4 [×2], C4⋊C8, C2.D8 [×2], C4×D4, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, D4⋊Q8, C12⋊C8, C6.Q16 [×2], D4⋊Dic3 [×2], C122Q8, D4×C12, C12.50D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], D8 [×2], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×D8, C8.C22, D4⋊S3 [×2], C2×Dic6, C4○D12, C2×C3⋊D4, D4⋊Q8, C12.48D4, C2×D4⋊S3, Q8.14D6, C12.50D8

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + + - - + - image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 D6 D8 C4○D4 C3⋊D4 Dic6 C4○D12 C8.C22 D4⋊S3 Q8.14D6 kernel C12.50D8 C12⋊C8 C6.Q16 D4⋊Dic3 C12⋊2Q8 D4×C12 C4×D4 C2×C12 C3×D4 C42 C4⋊C4 C2×D4 C12 C12 C2×C4 D4 C4 C6 C4 C2 # reps 1 1 2 2 1 1 1 2 2 1 1 1 4 2 4 4 4 1 2 2

Matrix representation of C12.50D8 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 3 0 0 0 33 49
,
 16 57 0 0 16 16 0 0 0 0 32 7 0 0 52 41
,
 16 57 0 0 57 57 0 0 0 0 32 7 0 0 10 41
`G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,3,33,0,0,0,49],[16,16,0,0,57,16,0,0,0,0,32,52,0,0,7,41],[16,57,0,0,57,57,0,0,0,0,32,10,0,0,7,41] >;`

C12.50D8 in GAP, Magma, Sage, TeX

`C_{12}._{50}D_8`
`% in TeX`

`G:=Group("C12.50D8");`
`// GroupNames label`

`G:=SmallGroup(192,566);`
`// by ID`

`G=gap.SmallGroup(192,566);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,1123,297,136,6278]);`
`// Polycyclic`

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

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