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## G = C32⋊D16order 288 = 25·32

### The semidirect product of C32 and D16 acting via D16/C4=D4

Aliases: C32⋊D16, C4.1S3≀C2, (C3×C6).1D8, (C3×C12).5D4, C322D81C2, C2.3(C32⋊D8), C322C161C2, C324C8.1C22, SmallGroup(288,382)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊4C8 — C32⋊D16
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2D8 — C32⋊D16
 Lower central C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊D16
 Upper central C1 — C2 — C4

Generators and relations for C32⋊D16
G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

24C2
24C2
2C3
2C3
12C22
12C22
2C6
2C6
8S3
8S3
24C6
24C6
6D4
6D4
9C8
2C12
2C12
4D6
4D6
12C2×C6
12C2×C6
9C16
9D8
9D8
2D12
2D12
9D16

Character table of C32⋊D16

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 16A 16B 16C 16D size 1 1 24 24 4 4 2 4 4 24 24 24 24 18 18 8 8 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 2 2 2 2 2 0 0 0 0 -2 -2 2 2 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 0 2 2 -2 2 2 0 0 0 0 0 0 -2 -2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ7 2 2 0 0 2 2 -2 2 2 0 0 0 0 0 0 -2 -2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ8 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 -√2 √2 0 0 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 orthogonal lifted from D16 ρ9 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 -√2 √2 0 0 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 orthogonal lifted from D16 ρ10 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 √2 -√2 0 0 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 orthogonal lifted from D16 ρ11 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 √2 -√2 0 0 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 orthogonal lifted from D16 ρ12 4 4 0 -2 1 -2 4 1 -2 0 1 1 0 0 0 -2 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 4 2 0 -2 1 4 -2 1 -1 0 0 -1 0 0 1 -2 0 0 0 0 orthogonal lifted from S3≀C2 ρ14 4 4 -2 0 -2 1 4 -2 1 1 0 0 1 0 0 1 -2 0 0 0 0 orthogonal lifted from S3≀C2 ρ15 4 4 0 2 1 -2 4 1 -2 0 -1 -1 0 0 0 -2 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 0 0 1 -2 -4 1 -2 0 √-3 -√-3 0 0 0 2 -1 0 0 0 0 complex lifted from C32⋊D8 ρ17 4 4 0 0 -2 1 -4 -2 1 -√-3 0 0 √-3 0 0 -1 2 0 0 0 0 complex lifted from C32⋊D8 ρ18 4 4 0 0 1 -2 -4 1 -2 0 -√-3 √-3 0 0 0 2 -1 0 0 0 0 complex lifted from C32⋊D8 ρ19 4 4 0 0 -2 1 -4 -2 1 √-3 0 0 -√-3 0 0 -1 2 0 0 0 0 complex lifted from C32⋊D8 ρ20 8 -8 0 0 2 -4 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2 ρ21 8 -8 0 0 -4 2 0 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of C32⋊D16
On 48 points
Generators in S48
```(1 21 48)(3 34 23)(5 25 36)(7 38 27)(9 29 40)(11 42 31)(13 17 44)(15 46 19)
(2 33 22)(4 24 35)(6 37 26)(8 28 39)(10 41 30)(12 32 43)(14 45 18)(16 20 47)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 36)(18 35)(19 34)(20 33)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)```

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

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

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

Matrix representation of C32⋊D16 in GL6(𝔽97)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 1 0 0 0 0 96 0 0 0 0 0 73 0 1 0 0 0 73 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 24 24 96 96
,
 95 71 0 0 0 0 26 95 0 0 0 0 0 0 0 0 1 96 0 0 73 73 2 1 0 0 34 34 24 0 0 0 34 35 24 0
,
 1 0 0 0 0 0 0 96 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 16 16 96 0 0 0 16 16 0 96

`G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,73,73,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,24,0,0,0,1,0,24,0,0,0,0,0,96,0,0,0,0,1,96],[95,26,0,0,0,0,71,95,0,0,0,0,0,0,0,73,34,34,0,0,0,73,34,35,0,0,1,2,24,24,0,0,96,1,0,0],[1,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,16,16,0,0,1,0,16,16,0,0,0,0,96,0,0,0,0,0,0,96] >;`

C32⋊D16 in GAP, Magma, Sage, TeX

`C_3^2\rtimes D_{16}`
`% in TeX`

`G:=Group("C3^2:D16");`
`// GroupNames label`

`G:=SmallGroup(288,382);`
`// by ID`

`G=gap.SmallGroup(288,382);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,254,135,142,675,346,80,2693,2028,691,797,2372]);`
`// Polycyclic`

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

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