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## G = C12⋊4D8order 192 = 26·3

### 1st semidirect product of C12 and D8 acting via D8/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C12⋊4D8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4⋊D12 — C12⋊4D8
 Lower central C3 — C6 — C2×C12 — C12⋊4D8
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C124D8
G = < a,b,c | a12=b8=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 728 in 162 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, D6, C2×C6, C42, C2×C8, D8, C2×D4, C24, D12, C2×C12, C2×C12, C22×S3, C4×C8, C41D4, C2×D8, D24, C4×C12, C2×C24, C2×D12, C2×D12, C84D4, C4×C24, C4⋊D12, C2×D24, C124D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, C41D4, C2×D8, D24, C2×D12, C84D4, C4⋊D12, C2×D24, C124D8

Smallest permutation representation of C124D8
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 63 76 15 46 95 36 51)(2 64 77 16 47 96 25 52)(3 65 78 17 48 85 26 53)(4 66 79 18 37 86 27 54)(5 67 80 19 38 87 28 55)(6 68 81 20 39 88 29 56)(7 69 82 21 40 89 30 57)(8 70 83 22 41 90 31 58)(9 71 84 23 42 91 32 59)(10 72 73 24 43 92 33 60)(11 61 74 13 44 93 34 49)(12 62 75 14 45 94 35 50)
(1 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 29)(14 28)(15 27)(16 26)(17 25)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 96)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 84)(59 83)(60 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), (1,63,76,15,46,95,36,51)(2,64,77,16,47,96,25,52)(3,65,78,17,48,85,26,53)(4,66,79,18,37,86,27,54)(5,67,80,19,38,87,28,55)(6,68,81,20,39,88,29,56)(7,69,82,21,40,89,30,57)(8,70,83,22,41,90,31,58)(9,71,84,23,42,91,32,59)(10,72,73,24,43,92,33,60)(11,61,74,13,44,93,34,49)(12,62,75,14,45,94,35,50), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,29)(14,28)(15,27)(16,26)(17,25)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,84)(59,83)(60,82)>;`

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

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

54 conjugacy classes

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

54 irreducible representations

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

Matrix representation of C124D8 in GL4(𝔽73) generated by

 59 66 0 0 7 66 0 0 0 0 0 1 0 0 72 72
,
 23 5 0 0 68 18 0 0 0 0 55 5 0 0 68 50
,
 50 68 0 0 18 23 0 0 0 0 50 68 0 0 18 23
`G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,0,72,0,0,1,72],[23,68,0,0,5,18,0,0,0,0,55,68,0,0,5,50],[50,18,0,0,68,23,0,0,0,0,50,18,0,0,68,23] >;`

C124D8 in GAP, Magma, Sage, TeX

`C_{12}\rtimes_4D_8`
`% in TeX`

`G:=Group("C12:4D8");`
`// GroupNames label`

`G:=SmallGroup(192,254);`
`// by ID`

`G=gap.SmallGroup(192,254);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,226,1123,136,6278]);`
`// Polycyclic`

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

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