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## G = C3⋊D48order 288 = 25·32

### The semidirect product of C3 and D48 acting via D48/D24=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C24 — C3⋊D48
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C3×D24 — C3⋊D48
 Lower central C32 — C3×C6 — C3×C12 — C3×C24 — C3⋊D48
 Upper central C1 — C2 — C4 — C8

Generators and relations for C3⋊D48
G = < a,b,c | a3=b48=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 498 in 67 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C12, C12, D6, C2×C6, C16, D8, C3×S3, C3⋊S3, C3×C6, C24, C24, D12, C3×D4, D16, C3×C12, S3×C6, C2×C3⋊S3, C3⋊C16, C48, D24, D24, C3×D8, C3×C24, C3×D12, C12⋊S3, D48, C3⋊D16, C3×C3⋊C16, C3×D24, C325D8, C3⋊D48
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, C3⋊D4, D16, S32, D24, D4⋊S3, C3⋊D12, D48, C3⋊D16, C3⋊D24, C3⋊D48

Smallest permutation representation of C3⋊D48
On 48 points
Generators in S48
```(1 33 17)(2 18 34)(3 35 19)(4 20 36)(5 37 21)(6 22 38)(7 39 23)(8 24 40)(9 41 25)(10 26 42)(11 43 27)(12 28 44)(13 45 29)(14 30 46)(15 47 31)(16 32 48)
(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)```

`G:=sub<Sym(48)| (1,33,17)(2,18,34)(3,35,19)(4,20,36)(5,37,21)(6,22,38)(7,39,23)(8,24,40)(9,41,25)(10,26,42)(11,43,27)(12,28,44)(13,45,29)(14,30,46)(15,47,31)(16,32,48), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)>;`

`G:=Group( (1,33,17)(2,18,34)(3,35,19)(4,20,36)(5,37,21)(6,22,38)(7,39,23)(8,24,40)(9,41,25)(10,26,42)(11,43,27)(12,28,44)(13,45,29)(14,30,46)(15,47,31)(16,32,48), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,48)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,40)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33) );`

`G=PermutationGroup([[(1,33,17),(2,18,34),(3,35,19),(4,20,36),(5,37,21),(6,22,38),(7,39,23),(8,24,40),(9,41,25),(10,26,42),(11,43,27),(12,28,44),(13,45,29),(14,30,46),(15,47,31),(16,32,48)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,48),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,40),(25,39),(26,38),(27,37),(28,36),(29,35),(30,34),(31,33)]])`

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 16A 16B 16C 16D 24A 24B 24C 24D 24E ··· 24J 48A ··· 48H order 1 2 2 2 3 3 3 4 6 6 6 6 6 8 8 12 12 12 12 12 16 16 16 16 24 24 24 24 24 ··· 24 48 ··· 48 size 1 1 24 72 2 2 4 2 2 2 4 24 24 2 2 2 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D4 D6 D8 D12 C3⋊D4 D16 D24 D48 S32 D4⋊S3 C3⋊D12 C3⋊D16 C3⋊D24 C3⋊D48 kernel C3⋊D48 C3×C3⋊C16 C3×D24 C32⋊5D8 C3⋊C16 D24 C3×C12 C24 C3×C6 C12 C12 C32 C6 C3 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 2 2 2 4 4 8 1 1 1 2 2 4

Matrix representation of C3⋊D48 in GL4(𝔽97) generated by

 0 96 0 0 1 96 0 0 0 0 1 0 0 0 0 1
,
 0 96 0 0 96 0 0 0 0 0 32 84 0 0 13 19
,
 0 1 0 0 1 0 0 0 0 0 95 18 0 0 16 2
`G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[0,96,0,0,96,0,0,0,0,0,32,13,0,0,84,19],[0,1,0,0,1,0,0,0,0,0,95,16,0,0,18,2] >;`

C3⋊D48 in GAP, Magma, Sage, TeX

`C_3\rtimes D_{48}`
`% in TeX`

`G:=Group("C3:D48");`
`// GroupNames label`

`G:=SmallGroup(288,194);`
`// by ID`

`G=gap.SmallGroup(288,194);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,254,142,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^48=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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