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## G = C3⋊S3⋊3C16order 288 = 25·32

### 2nd semidirect product of C3⋊S3 and C16 acting via C16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3⋊3C16
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2C16 — C3⋊S3⋊3C16
 Lower central C32 — C3⋊S3⋊3C16
 Upper central C1 — C8

Generators and relations for C3⋊S33C16
G = < a,b,c,d | a3=b3=c2=d16=1, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a-1b-1, cd=dc >

Smallest permutation representation of C3⋊S33C16
On 48 points
Generators in S48
```(1 25 39)(2 26 40)(3 41 27)(4 42 28)(5 29 43)(6 30 44)(7 45 31)(8 46 32)(9 17 47)(10 18 48)(11 33 19)(12 34 20)(13 21 35)(14 22 36)(15 37 23)(16 38 24)
(2 40 26)(4 28 42)(6 44 30)(8 32 46)(10 48 18)(12 20 34)(14 36 22)(16 24 38)
(17 47)(18 48)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)
(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)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 16A ··· 16P 24A ··· 24H order 1 2 2 2 3 3 4 4 4 4 6 6 8 8 8 8 8 8 8 8 12 12 12 12 16 ··· 16 24 ··· 24 size 1 1 9 9 4 4 1 1 9 9 4 4 1 1 1 1 9 9 9 9 4 4 4 4 9 ··· 9 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C8 C8 C16 C32⋊C4 C2×C32⋊C4 C3⋊S3⋊3C8 C3⋊S3⋊3C16 kernel C3⋊S3⋊3C16 C32⋊2C16 C8×C3⋊S3 C3×C24 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3⋊S3 C8 C4 C2 C1 # reps 1 2 1 2 2 4 4 16 2 2 4 8

Matrix representation of C3⋊S33C16 in GL4(𝔽97) generated by

 0 1 0 64 96 96 33 0 0 0 0 96 0 0 1 96
,
 1 0 64 0 0 1 64 0 0 0 96 1 0 0 96 0
,
 0 1 0 64 1 0 0 64 0 0 1 96 0 0 0 96
,
 22 0 81 82 22 0 81 81 66 33 75 75 33 64 0 0
`G:=sub<GL(4,GF(97))| [0,96,0,0,1,96,0,0,0,33,0,1,64,0,96,96],[1,0,0,0,0,1,0,0,64,64,96,96,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,64,64,96,96],[22,22,66,33,0,0,33,64,81,81,75,0,82,81,75,0] >;`

C3⋊S33C16 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3\rtimes_3C_{16}`
`% in TeX`

`G:=Group("C3:S3:3C16");`
`// GroupNames label`

`G:=SmallGroup(288,412);`
`// by ID`

`G=gap.SmallGroup(288,412);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,58,80,9413,691,12550,2372]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^2=d^16=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,c*d=d*c>;`
`// generators/relations`

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