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## G = C4.5D24order 192 = 26·3

### 5th non-split extension by C4 of D24 acting via D24/C24=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C4.5D24
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4⋊D12 — C4.5D24
 Lower central C3 — C6 — C2×C12 — C4.5D24
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C4.5D24
G = < a,b,c | a4=b24=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

Subgroups: 472 in 118 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C4×C8, D4⋊C4, C41D4, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×D12, C2×D12, C4.4D8, C2.D24, C4×C24, C122Q8, C4⋊D12, C4.5D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, SD16, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C2×D8, C2×SD16, C24⋊C2, D24, C2×D12, C4○D12, C4.4D8, C427S3, C2×C24⋊C2, C2×D24, C4.5D24

Smallest permutation representation of C4.5D24
On 96 points
Generators in S96
```(1 95 25 59)(2 96 26 60)(3 73 27 61)(4 74 28 62)(5 75 29 63)(6 76 30 64)(7 77 31 65)(8 78 32 66)(9 79 33 67)(10 80 34 68)(11 81 35 69)(12 82 36 70)(13 83 37 71)(14 84 38 72)(15 85 39 49)(16 86 40 50)(17 87 41 51)(18 88 42 52)(19 89 43 53)(20 90 44 54)(21 91 45 55)(22 92 46 56)(23 93 47 57)(24 94 48 58)
(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 48 25 24)(2 23 26 47)(3 46 27 22)(4 21 28 45)(5 44 29 20)(6 19 30 43)(7 42 31 18)(8 17 32 41)(9 40 33 16)(10 15 34 39)(11 38 35 14)(12 13 36 37)(49 68 85 80)(50 79 86 67)(51 66 87 78)(52 77 88 65)(53 64 89 76)(54 75 90 63)(55 62 91 74)(56 73 92 61)(57 60 93 96)(58 95 94 59)(69 72 81 84)(70 83 82 71)```

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

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

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

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 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D6 D8 SD16 C4○D4 D12 C24⋊C2 D24 C4○D12 kernel C4.5D24 C2.D24 C4×C24 C12⋊2Q8 C4⋊D12 C4×C8 C2×C12 C42 C2×C8 C12 C12 C12 C2×C4 C4 C4 C4 # reps 1 4 1 1 1 1 2 1 2 4 4 4 4 8 8 8

Matrix representation of C4.5D24 in GL4(𝔽73) generated by

 66 59 0 0 14 7 0 0 0 0 1 0 0 0 0 1
,
 62 37 0 0 36 25 0 0 0 0 68 50 0 0 23 18
,
 62 37 0 0 48 11 0 0 0 0 68 50 0 0 55 5
`G:=sub<GL(4,GF(73))| [66,14,0,0,59,7,0,0,0,0,1,0,0,0,0,1],[62,36,0,0,37,25,0,0,0,0,68,23,0,0,50,18],[62,48,0,0,37,11,0,0,0,0,68,55,0,0,50,5] >;`

C4.5D24 in GAP, Magma, Sage, TeX

`C_4._5D_{24}`
`% in TeX`

`G:=Group("C4.5D24");`
`// GroupNames label`

`G:=SmallGroup(192,253);`
`// by ID`

`G=gap.SmallGroup(192,253);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,142,1123,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^24=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;`
`// generators/relations`

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