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## G = C2.D48order 192 = 26·3

### 2nd central extension by C2 of D48

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C2.D48
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C2×D24 — C2.D48
 Lower central C3 — C6 — C12 — C24 — C2.D48
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2.D48
G = < a,b,c | a2=b48=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >

Subgroups: 328 in 66 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, D6, C2×C6, C16, C4⋊C4, C2×C8, D8, C2×D4, C24, D12, C2×Dic3, C2×C12, C22×S3, C2.D8, C2×C16, C2×D8, C48, D24, D24, C4⋊Dic3, C2×C24, C2×D12, C2.D16, C241C4, C2×C48, C2×D24, C2.D48
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, D16, SD32, C24⋊C2, D24, D6⋊C4, C2.D16, D48, C48⋊C2, C2.D24, C2.D48

Smallest permutation representation of C2.D48
On 96 points
Generators in S96
```(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)
(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 74 75 48)(2 47 76 73)(3 72 77 46)(4 45 78 71)(5 70 79 44)(6 43 80 69)(7 68 81 42)(8 41 82 67)(9 66 83 40)(10 39 84 65)(11 64 85 38)(12 37 86 63)(13 62 87 36)(14 35 88 61)(15 60 89 34)(16 33 90 59)(17 58 91 32)(18 31 92 57)(19 56 93 30)(20 29 94 55)(21 54 95 28)(22 27 96 53)(23 52 49 26)(24 25 50 51)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 16A ··· 16H 24A ··· 24H 48A ··· 48P order 1 2 2 2 2 2 3 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 24 24 2 2 2 24 24 2 2 2 2 2 2 2 2 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 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D4 D6 SD16 D8 C4×S3 C3⋊D4 D12 D16 SD32 C24⋊C2 D24 D48 C48⋊C2 kernel C2.D48 C24⋊1C4 C2×C48 C2×D24 D24 C2×C16 C24 C2×C12 C2×C8 C12 C2×C6 C8 C8 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 4 4 4 4 8 8

Matrix representation of C2.D48 in GL3(𝔽97) generated by

 96 0 0 0 96 0 0 0 96
,
 75 0 0 0 43 45 0 52 95
,
 22 0 0 0 43 45 0 2 54
`G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[75,0,0,0,43,52,0,45,95],[22,0,0,0,43,2,0,45,54] >;`

C2.D48 in GAP, Magma, Sage, TeX

`C_2.D_{48}`
`% in TeX`

`G:=Group("C2.D48");`
`// GroupNames label`

`G:=SmallGroup(192,68);`
`// by ID`

`G=gap.SmallGroup(192,68);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,204,422,268,1684,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^2=b^48=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;`
`// generators/relations`

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