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## G = C3⋊S3⋊Q16order 288 = 25·32

### The semidirect product of C3⋊S3 and Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C3⋊S3⋊Q16
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2Q8 — C32⋊Q16 — C3⋊S3⋊Q16
 Lower central C32 — C3×C6 — C3⋊Dic3 — C3⋊S3⋊Q16
 Upper central C1 — C2 — C4

Generators and relations for C3⋊S3⋊Q16
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=d4, ab=ba, cac=dbd-1=ebe-1=a-1, dad-1=b, eae-1=cbc=b-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 464 in 98 conjugacy classes, 25 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×5], C22, S3 [×4], C6 [×2], C8 [×2], C2×C4 [×3], Q8 [×6], C32, Dic3 [×6], C12 [×6], D6 [×2], C2×C8, Q16 [×4], C2×Q8 [×2], C3⋊S3 [×2], C3×C6, Dic6 [×6], C4×S3 [×6], C3×Q8 [×2], C2×Q16, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8 [×2], C322C8 [×2], C6.D6 [×2], C322Q8 [×4], C3×Dic6 [×2], C4×C3⋊S3, C32⋊Q16 [×4], C3⋊S33C8, Dic3.D6 [×2], C3⋊S3⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, Q16 [×2], C2×D4, C2×Q16, S3≀C2, C2×S3≀C2, C3⋊S3⋊Q16

Character table of C3⋊S3⋊Q16

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 9 9 4 4 2 12 12 12 12 18 4 4 18 18 18 18 8 8 24 24 24 24 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 -2 -2 2 2 2 0 0 0 0 -2 2 2 0 0 0 0 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -2 0 0 0 0 -2 2 2 0 0 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -√2 √2 √2 -√2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 √2 √2 -√2 -√2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 √2 -√2 -√2 √2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 -√2 -√2 √2 √2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 4 4 0 0 -2 1 -4 0 0 -2 2 0 1 -2 0 0 0 0 2 -1 0 0 -1 1 orthogonal lifted from C2×S3≀C2 ρ16 4 4 0 0 -2 1 4 0 0 2 2 0 1 -2 0 0 0 0 -2 1 0 0 -1 -1 orthogonal lifted from S3≀C2 ρ17 4 4 0 0 1 -2 4 -2 -2 0 0 0 -2 1 0 0 0 0 1 -2 1 1 0 0 orthogonal lifted from S3≀C2 ρ18 4 4 0 0 1 -2 -4 -2 2 0 0 0 -2 1 0 0 0 0 -1 2 1 -1 0 0 orthogonal lifted from C2×S3≀C2 ρ19 4 4 0 0 1 -2 -4 2 -2 0 0 0 -2 1 0 0 0 0 -1 2 -1 1 0 0 orthogonal lifted from C2×S3≀C2 ρ20 4 4 0 0 1 -2 4 2 2 0 0 0 -2 1 0 0 0 0 1 -2 -1 -1 0 0 orthogonal lifted from S3≀C2 ρ21 4 4 0 0 -2 1 -4 0 0 2 -2 0 1 -2 0 0 0 0 2 -1 0 0 1 -1 orthogonal lifted from C2×S3≀C2 ρ22 4 4 0 0 -2 1 4 0 0 -2 -2 0 1 -2 0 0 0 0 -2 1 0 0 1 1 orthogonal lifted from S3≀C2 ρ23 8 -8 0 0 -4 2 0 0 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ24 8 -8 0 0 2 -4 0 0 0 0 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C3⋊S3⋊Q16
On 48 points
Generators in S48
```(1 46 11)(3 13 48)(5 42 15)(7 9 44)(18 29 35)(20 37 31)(22 25 39)(24 33 27)
(2 12 47)(4 41 14)(6 16 43)(8 45 10)(17 28 34)(19 36 30)(21 32 38)(23 40 26)
(1 5)(2 6)(3 7)(4 8)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 21)(18 22)(19 23)(20 24)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)
(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)| (1,46,11)(3,13,48)(5,42,15)(7,9,44)(18,29,35)(20,37,31)(22,25,39)(24,33,27), (2,12,47)(4,41,14)(6,16,43)(8,45,10)(17,28,34)(19,36,30)(21,32,38)(23,40,26), (1,5)(2,6)(3,7)(4,8)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,21)(18,22)(19,23)(20,24)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34), (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( (1,46,11)(3,13,48)(5,42,15)(7,9,44)(18,29,35)(20,37,31)(22,25,39)(24,33,27), (2,12,47)(4,41,14)(6,16,43)(8,45,10)(17,28,34)(19,36,30)(21,32,38)(23,40,26), (1,5)(2,6)(3,7)(4,8)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,21)(18,22)(19,23)(20,24)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34), (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([(1,46,11),(3,13,48),(5,42,15),(7,9,44),(18,29,35),(20,37,31),(22,25,39),(24,33,27)], [(2,12,47),(4,41,14),(6,16,43),(8,45,10),(17,28,34),(19,36,30),(21,32,38),(23,40,26)], [(1,5),(2,6),(3,7),(4,8),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,21),(18,22),(19,23),(20,24),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,33),(32,34)], [(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 C3⋊S3⋊Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 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 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 38 0 0 0 0 48 41 0 0 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 72 0 0 0 0 0 0 72 0 0
,
 24 70 0 0 0 0 22 49 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,48,0,0,0,0,38,41,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,72,0,0,0],[24,22,0,0,0,0,70,49,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C3⋊S3⋊Q16 in GAP, Magma, Sage, TeX

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

`G:=Group("C3:S3:Q16");`
`// GroupNames label`

`G:=SmallGroup(288,876);`
`// by ID`

`G=gap.SmallGroup(288,876);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,141,120,422,219,100,675,346,80,2693,2028,362,797,1203]);`
`// Polycyclic`

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

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