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## G = C33⋊Q16order 432 = 24·33

### 2nd semidirect product of C33 and Q16 acting via Q16/C2=D4

Aliases: C332Q16, C6.18S3≀C2, C334C8.C2, C322Q8.S3, C32(C32⋊Q16), C3⋊Dic3.13D6, (C32×C6).12D4, C335Q8.2C2, C323(C3⋊Q16), C2.7(C33⋊D4), (C3×C6).18(C3⋊D4), (C3×C322Q8).1C2, (C3×C3⋊Dic3).10C22, SmallGroup(432,585)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊Dic3 — C33⋊Q16
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊5Q8 — C33⋊Q16
 Lower central C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊Q16
 Upper central C1 — C2

Generators and relations for C33⋊Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, dad-1=b-1, eae-1=a-1, bc=cb, dbd-1=a, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 412 in 72 conjugacy classes, 15 normal (all characteristic)
C1, C2, C3, C3 [×4], C4 [×3], C6, C6 [×4], C8, Q8 [×2], C32, C32 [×4], Dic3 [×6], C12 [×5], Q16, C3×C6, C3×C6 [×4], C3⋊C8, Dic6 [×3], C3×Q8, C33, C3×Dic3 [×8], C3⋊Dic3, C3⋊Dic3, C3×C12, C3⋊Q16, C32×C6, C322C8, C322Q8, C322Q8 [×2], C3×Dic6, C32×Dic3, C3×C3⋊Dic3, C3×C3⋊Dic3, C32⋊Q16, C334C8, C3×C322Q8, C335Q8, C33⋊Q16
Quotients: C1, C2 [×3], C22, S3, D4, D6, Q16, C3⋊D4, C3⋊Q16, S3≀C2, C32⋊Q16, C33⋊D4, C33⋊Q16

Character table of C33⋊Q16

 class 1 2 3A 3B 3C 3D 3E 3F 4A 4B 4C 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K size 1 1 2 4 4 4 4 8 12 18 36 2 4 4 4 4 8 54 54 12 12 12 12 12 12 12 12 36 36 36 ρ1 1 1 1 1 1 1 1 1 1 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 -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 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 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ5 2 2 2 2 2 2 2 2 0 -2 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 -2 0 orthogonal lifted from D4 ρ6 2 2 -1 2 -1 2 -1 -1 2 2 0 -1 -1 2 2 -1 -1 0 0 2 -1 -1 2 -1 -1 -1 -1 0 -1 0 orthogonal lifted from S3 ρ7 2 2 -1 2 -1 2 -1 -1 -2 2 0 -1 -1 2 2 -1 -1 0 0 -2 1 1 -2 1 1 1 1 0 -1 0 orthogonal lifted from D6 ρ8 2 -2 2 2 2 2 2 2 0 0 0 -2 -2 -2 -2 -2 -2 √2 -√2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ9 2 -2 2 2 2 2 2 2 0 0 0 -2 -2 -2 -2 -2 -2 -√2 √2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ10 2 2 -1 2 -1 2 -1 -1 0 -2 0 -1 -1 2 2 -1 -1 0 0 0 -√-3 √-3 0 -√-3 √-3 -√-3 √-3 0 1 0 complex lifted from C3⋊D4 ρ11 2 2 -1 2 -1 2 -1 -1 0 -2 0 -1 -1 2 2 -1 -1 0 0 0 √-3 -√-3 0 √-3 -√-3 √-3 -√-3 0 1 0 complex lifted from C3⋊D4 ρ12 4 4 4 -2 1 1 1 -2 -2 0 0 4 1 -2 1 1 -2 0 0 1 1 1 1 1 1 -2 -2 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 4 4 -2 1 1 1 -2 2 0 0 4 1 -2 1 1 -2 0 0 -1 -1 -1 -1 -1 -1 2 2 0 0 0 orthogonal lifted from S3≀C2 ρ14 4 4 4 1 -2 -2 -2 1 0 0 2 4 -2 1 -2 -2 1 0 0 0 0 0 0 0 0 0 0 -1 0 -1 orthogonal lifted from S3≀C2 ρ15 4 4 4 1 -2 -2 -2 1 0 0 -2 4 -2 1 -2 -2 1 0 0 0 0 0 0 0 0 0 0 1 0 1 orthogonal lifted from S3≀C2 ρ16 4 -4 -2 4 -2 4 -2 -2 0 0 0 2 2 -4 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C3⋊Q16, Schur index 2 ρ17 4 -4 4 -2 1 1 1 -2 0 0 0 -4 -1 2 -1 -1 2 0 0 √3 √3 √3 -√3 -√3 -√3 0 0 0 0 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ18 4 -4 4 -2 1 1 1 -2 0 0 0 -4 -1 2 -1 -1 2 0 0 -√3 -√3 -√3 √3 √3 √3 0 0 0 0 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ19 4 -4 4 1 -2 -2 -2 1 0 0 0 -4 2 -1 2 2 -1 0 0 0 0 0 0 0 0 0 0 -√3 0 √3 symplectic lifted from C32⋊Q16, Schur index 2 ρ20 4 -4 4 1 -2 -2 -2 1 0 0 0 -4 2 -1 2 2 -1 0 0 0 0 0 0 0 0 0 0 √3 0 -√3 symplectic lifted from C32⋊Q16, Schur index 2 ρ21 4 4 -2 -2 -1-3√-3/2 1 -1+3√-3/2 1 -2 0 0 -2 -1-3√-3/2 -2 1 -1+3√-3/2 1 0 0 1 ζ3 ζ32 1 ζ3 ζ32 1-√-3 1+√-3 0 0 0 complex lifted from C33⋊D4 ρ22 4 4 -2 -2 -1+3√-3/2 1 -1-3√-3/2 1 -2 0 0 -2 -1+3√-3/2 -2 1 -1-3√-3/2 1 0 0 1 ζ32 ζ3 1 ζ32 ζ3 1+√-3 1-√-3 0 0 0 complex lifted from C33⋊D4 ρ23 4 -4 -2 -2 -1+3√-3/2 1 -1-3√-3/2 1 0 0 0 2 1-3√-3/2 2 -1 1+3√-3/2 -1 0 0 -√3 ζ4ζ32+2ζ4 ζ43ζ3+2ζ43 √3 ζ43ζ32+2ζ43 ζ4ζ3+2ζ4 0 0 0 0 0 complex faithful ρ24 4 4 -2 -2 -1-3√-3/2 1 -1+3√-3/2 1 2 0 0 -2 -1-3√-3/2 -2 1 -1+3√-3/2 1 0 0 -1 ζ65 ζ6 -1 ζ65 ζ6 -1+√-3 -1-√-3 0 0 0 complex lifted from C33⋊D4 ρ25 4 -4 -2 -2 -1-3√-3/2 1 -1+3√-3/2 1 0 0 0 2 1+3√-3/2 2 -1 1-3√-3/2 -1 0 0 -√3 ζ43ζ3+2ζ43 ζ4ζ32+2ζ4 √3 ζ4ζ3+2ζ4 ζ43ζ32+2ζ43 0 0 0 0 0 complex faithful ρ26 4 4 -2 -2 -1+3√-3/2 1 -1-3√-3/2 1 2 0 0 -2 -1+3√-3/2 -2 1 -1-3√-3/2 1 0 0 -1 ζ6 ζ65 -1 ζ6 ζ65 -1-√-3 -1+√-3 0 0 0 complex lifted from C33⋊D4 ρ27 4 -4 -2 -2 -1+3√-3/2 1 -1-3√-3/2 1 0 0 0 2 1-3√-3/2 2 -1 1+3√-3/2 -1 0 0 √3 ζ43ζ32+2ζ43 ζ4ζ3+2ζ4 -√3 ζ4ζ32+2ζ4 ζ43ζ3+2ζ43 0 0 0 0 0 complex faithful ρ28 4 -4 -2 -2 -1-3√-3/2 1 -1+3√-3/2 1 0 0 0 2 1+3√-3/2 2 -1 1-3√-3/2 -1 0 0 √3 ζ4ζ3+2ζ4 ζ43ζ32+2ζ43 -√3 ζ43ζ3+2ζ43 ζ4ζ32+2ζ4 0 0 0 0 0 complex faithful ρ29 8 8 -4 2 2 -4 2 -1 0 0 0 -4 2 2 -4 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C33⋊D4 ρ30 8 -8 -4 2 2 -4 2 -1 0 0 0 4 -2 -2 4 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C33⋊Q16
On 48 points
Generators in S48
```(1 11 46)(2 47 12)(3 48 13)(4 14 41)(5 15 42)(6 43 16)(7 44 9)(8 10 45)(17 34 28)(18 35 29)(19 30 36)(20 31 37)(21 38 32)(22 39 25)(23 26 40)(24 27 33)
(1 11 46)(2 12 47)(3 48 13)(4 41 14)(5 15 42)(6 16 43)(7 44 9)(8 45 10)(17 28 34)(18 35 29)(19 36 30)(20 31 37)(21 32 38)(22 39 25)(23 40 26)(24 27 33)
(1 46 11)(2 12 47)(3 48 13)(4 14 41)(5 42 15)(6 16 43)(7 44 9)(8 10 45)(17 28 34)(18 35 29)(19 30 36)(20 37 31)(21 32 38)(22 39 25)(23 26 40)(24 33 27)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 34 13 38)(10 33 14 37)(11 40 15 36)(12 39 16 35)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)```

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

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

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

Matrix representation of C33⋊Q16 in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 1 0 0 72
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 72
,
 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 0 41 0 0 0 0 0 0 41 0 0 0 0 0 0 16 0 41 0 0 0 0 16 0 41 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0
,
 41 0 13 0 0 0 0 0 0 41 0 13 0 0 0 0 11 0 32 0 0 0 0 0 0 11 0 32 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,72,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,72],[0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,41,0,41,0,0,0,0,41,0,41,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0],[41,0,11,0,0,0,0,0,0,41,0,11,0,0,0,0,13,0,32,0,0,0,0,0,0,13,0,32,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C33⋊Q16 in GAP, Magma, Sage, TeX

`C_3^3\rtimes Q_{16}`
`% in TeX`

`G:=Group("C3^3:Q16");`
`// GroupNames label`

`G:=SmallGroup(432,585);`
`// by ID`

`G=gap.SmallGroup(432,585);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,254,135,58,1684,571,298,677,1027,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,d*a*d^-1=b^-1,e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=a,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;`
`// generators/relations`

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