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## G = D48⋊5C2order 192 = 26·3

### 5th semidirect product of D48 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D48⋊5C2
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — D24⋊C2 — D48⋊5C2
 Lower central C3 — C6 — C12 — C24 — D48⋊5C2
 Upper central C1 — C2 — C4 — C8 — Q32

Generators and relations for D485C2
G = < a,b,c | a48=b2=c2=1, bab=a-1, cac=a17, cbc=a40b >

Subgroups: 316 in 84 conjugacy classes, 31 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Dic3, C12, C12, D6, D6, C16, C16, C2×C8, D8, SD16, Q16, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12, C3×Q8, C2×C16, D16, SD32, Q32, C4○D8, C3⋊C16, C48, S3×C8, D24, Q82S3, C3×Q16, Q83S3, C4○D16, S3×C16, D48, C8.6D6, C3×Q32, D24⋊C2, D485C2
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, C2×D8, S3×D4, C4○D16, S3×D8, D485C2

Smallest permutation representation of D485C2
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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(49 56)(50 55)(51 54)(52 53)(57 96)(58 95)(59 94)(60 93)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)
(1 65)(2 82)(3 51)(4 68)(5 85)(6 54)(7 71)(8 88)(9 57)(10 74)(11 91)(12 60)(13 77)(14 94)(15 63)(16 80)(17 49)(18 66)(19 83)(20 52)(21 69)(22 86)(23 55)(24 72)(25 89)(26 58)(27 75)(28 92)(29 61)(30 78)(31 95)(32 64)(33 81)(34 50)(35 67)(36 84)(37 53)(38 70)(39 87)(40 56)(41 73)(42 90)(43 59)(44 76)(45 93)(46 62)(47 79)(48 96)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6 8A 8B 8C 8D 12A 12B 12C 16A 16B 16C 16D 16E 16F 16G 16H 24A 24B 48A 48B 48C 48D order 1 2 2 2 2 3 4 4 4 4 4 6 8 8 8 8 12 12 12 16 16 16 16 16 16 16 16 24 24 48 48 48 48 size 1 1 6 24 24 2 2 3 3 8 8 2 2 2 6 6 4 16 16 2 2 2 2 6 6 6 6 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 D8 C4○D16 S3×D4 S3×D8 D48⋊5C2 kernel D48⋊5C2 S3×C16 D48 C8.6D6 C3×Q32 D24⋊C2 Q32 C3⋊C8 C4×S3 C16 Q16 Dic3 D6 C3 C4 C2 C1 # reps 1 1 1 2 1 2 1 1 1 1 2 2 2 8 1 2 4

Matrix representation of D485C2 in GL4(𝔽97) generated by

 26 95 0 0 2 26 0 0 0 0 0 96 0 0 1 96
,
 26 95 0 0 95 71 0 0 0 0 1 96 0 0 0 96
,
 0 22 0 0 75 0 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(97))| [26,2,0,0,95,26,0,0,0,0,0,1,0,0,96,96],[26,95,0,0,95,71,0,0,0,0,1,0,0,0,96,96],[0,75,0,0,22,0,0,0,0,0,0,1,0,0,1,0] >;`

D485C2 in GAP, Magma, Sage, TeX

`D_{48}\rtimes_5C_2`
`% in TeX`

`G:=Group("D48:5C2");`
`// GroupNames label`

`G:=SmallGroup(192,478);`
`// by ID`

`G=gap.SmallGroup(192,478);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,758,135,184,346,185,192,851,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^48=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^17,c*b*c=a^40*b>;`
`// generators/relations`

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