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## G = C3⋊D16order 96 = 25·3

### The semidirect product of C3 and D16 acting via D16/D8=C2

Aliases: C32D16, D81S3, C6.8D8, C8.4D6, D243C2, C12.3D4, C24.2C22, C3⋊C161C2, (C3×D8)⋊1C2, C2.4(D4⋊S3), C4.1(C3⋊D4), SmallGroup(96,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3⋊D16
 Chief series C1 — C3 — C6 — C12 — C24 — D24 — C3⋊D16
 Lower central C3 — C6 — C12 — C24 — C3⋊D16
 Upper central C1 — C2 — C4 — C8 — D8

Generators and relations for C3⋊D16
G = < a,b,c | a3=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C3⋊D16

 class 1 2A 2B 2C 3 4 6A 6B 6C 8A 8B 12 16A 16B 16C 16D 24A 24B size 1 1 8 24 2 2 2 8 8 2 2 4 6 6 6 6 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 -2 0 -1 2 -1 1 1 2 2 -1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ6 2 2 2 0 -1 2 -1 -1 -1 2 2 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ7 2 2 0 0 2 2 2 0 0 -2 -2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ8 2 -2 0 0 2 0 -2 0 0 √2 -√2 0 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 √2 -√2 orthogonal lifted from D16 ρ9 2 -2 0 0 2 0 -2 0 0 -√2 √2 0 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 -√2 √2 orthogonal lifted from D16 ρ10 2 2 0 0 2 -2 2 0 0 0 0 -2 -√2 √2 -√2 √2 0 0 orthogonal lifted from D8 ρ11 2 2 0 0 2 -2 2 0 0 0 0 -2 √2 -√2 √2 -√2 0 0 orthogonal lifted from D8 ρ12 2 -2 0 0 2 0 -2 0 0 √2 -√2 0 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 √2 -√2 orthogonal lifted from D16 ρ13 2 -2 0 0 2 0 -2 0 0 -√2 √2 0 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -√2 √2 orthogonal lifted from D16 ρ14 2 2 0 0 -1 2 -1 -√-3 √-3 -2 -2 -1 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ15 2 2 0 0 -1 2 -1 √-3 -√-3 -2 -2 -1 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ16 4 4 0 0 -2 -4 -2 0 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ17 4 -4 0 0 -2 0 2 0 0 2√2 -2√2 0 0 0 0 0 -√2 √2 orthogonal faithful, Schur index 2 ρ18 4 -4 0 0 -2 0 2 0 0 -2√2 2√2 0 0 0 0 0 √2 -√2 orthogonal faithful, Schur index 2

Smallest permutation representation of C3⋊D16
On 48 points
Generators in S48
```(1 34 22)(2 23 35)(3 36 24)(4 25 37)(5 38 26)(6 27 39)(7 40 28)(8 29 41)(9 42 30)(10 31 43)(11 44 32)(12 17 45)(13 46 18)(14 19 47)(15 48 20)(16 21 33)
(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 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)```

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

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

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

C3⋊D16 is a maximal subgroup of
S3×D16  D8⋊D6  D48⋊C2  D6.2D8  D8.D6  Q16⋊D6  Q16.D6  C9⋊D16  C322D16  C3⋊D48  C327D16  C15⋊D16  C3⋊D80  C157D16
C3⋊D16 is a maximal quotient of
C6.6D16  C6.D16  C3⋊D32  D16.S3  C3⋊SD64  C3⋊Q64  D81Dic3  C9⋊D16  C322D16  C3⋊D48  C327D16  C15⋊D16  C3⋊D80  C157D16

Matrix representation of C3⋊D16 in GL4(𝔽7) generated by

 5 6 3 6 1 3 3 3 4 3 2 6 0 0 0 2
,
 3 5 2 3 3 1 1 5 5 3 2 0 2 0 3 1
,
 2 2 3 2 6 5 3 5 1 1 3 2 5 2 0 4
`G:=sub<GL(4,GF(7))| [5,1,4,0,6,3,3,0,3,3,2,0,6,3,6,2],[3,3,5,2,5,1,3,0,2,1,2,3,3,5,0,1],[2,6,1,5,2,5,1,2,3,3,3,0,2,5,2,4] >;`

C3⋊D16 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(96,33);`
`// by ID`

`G=gap.SmallGroup(96,33);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,218,116,122,579,297,69,2309]);`
`// Polycyclic`

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

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