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## G = C3⋊S3.5Q16order 288 = 25·32

### The non-split extension by C3⋊S3 of Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3⋊S3.5Q16
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C4⋊(C32⋊C4) — C3⋊S3.5Q16
 Lower central C32 — C3×C6 — C3×C12 — C3⋊S3.5Q16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for C3⋊S3.5Q16
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=d4, ab=ba, cac=a-1, dad-1=ab-1, ae=ea, cbc=b-1, dbd-1=a-1b-1, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 432 in 80 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×4], C22, S3 [×4], C6 [×2], C8, C2×C4 [×3], Q8, Q8 [×2], C32, Dic3 [×6], C12 [×6], D6 [×2], C4⋊C4, C2×C8, C2×Q8, C3⋊S3 [×2], C3×C6, Dic6 [×6], C4×S3 [×6], C3×Q8 [×2], Q8⋊C4, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C32⋊C4, C2×C3⋊S3, S3×Q8 [×2], C322C8, C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, C2×C32⋊C4, C3⋊S33C8, C4⋊(C32⋊C4), Q8×C3⋊S3, C3⋊S3.5Q16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, Q8⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C3⋊S3.5Q16

Character table of C3⋊S3.5Q16

 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 4 18 36 36 36 4 4 18 18 18 18 8 8 8 8 8 8 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 i -1 -i 1 1 i -i -i i 1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 -1 1 1 1 -1 -1 i 1 -i 1 1 -i i i -i -1 -1 -1 1 1 -1 linear of order 4 ρ7 1 1 -1 -1 1 1 1 -1 -1 -i 1 i 1 1 i -i -i i -1 -1 -1 1 1 -1 linear of order 4 ρ8 1 1 -1 -1 1 1 1 1 -1 -i -1 i 1 1 -i i i -i 1 1 1 1 1 1 linear of order 4 ρ9 2 2 -2 -2 2 2 -2 0 2 0 0 0 2 2 0 0 0 0 0 0 0 -2 -2 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -2 0 -2 0 0 0 2 2 0 0 0 0 0 0 0 -2 -2 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 complex lifted from SD16 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 complex lifted from SD16 ρ15 4 4 0 0 1 -2 4 4 0 0 0 0 -2 1 0 0 0 0 -2 1 1 1 -2 -2 orthogonal lifted from C32⋊C4 ρ16 4 4 0 0 1 -2 -4 0 0 0 0 0 -2 1 0 0 0 0 0 3 -3 -1 2 0 orthogonal lifted from C62⋊C4 ρ17 4 4 0 0 1 -2 -4 0 0 0 0 0 -2 1 0 0 0 0 0 -3 3 -1 2 0 orthogonal lifted from C62⋊C4 ρ18 4 4 0 0 -2 1 4 -4 0 0 0 0 1 -2 0 0 0 0 -1 2 2 -2 1 -1 orthogonal lifted from C2×C32⋊C4 ρ19 4 4 0 0 -2 1 -4 0 0 0 0 0 1 -2 0 0 0 0 3 0 0 2 -1 -3 orthogonal lifted from C62⋊C4 ρ20 4 4 0 0 -2 1 4 4 0 0 0 0 1 -2 0 0 0 0 1 -2 -2 -2 1 1 orthogonal lifted from C32⋊C4 ρ21 4 4 0 0 -2 1 -4 0 0 0 0 0 1 -2 0 0 0 0 -3 0 0 2 -1 3 orthogonal lifted from C62⋊C4 ρ22 4 4 0 0 1 -2 4 -4 0 0 0 0 -2 1 0 0 0 0 2 -1 -1 1 -2 2 orthogonal lifted from C2×C32⋊C4 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⋊S3.5Q16
On 48 points
Generators in S48
```(1 30 46)(2 31 47)(3 48 32)(4 41 25)(5 26 42)(6 27 43)(7 44 28)(8 45 29)(9 35 19)(10 36 20)(11 21 37)(12 22 38)(13 39 23)(14 40 24)(15 17 33)(16 18 34)
(2 47 31)(4 25 41)(6 43 27)(8 29 45)(10 20 36)(12 38 22)(14 24 40)(16 34 18)
(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 30 13 26)(10 45 14 41)(11 28 15 32)(12 43 16 47)(25 36 29 40)(27 34 31 38)(33 48 37 44)(35 46 39 42)```

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

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

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

Matrix representation of C3⋊S3.5Q16 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 1 72 72 0 0 72 0 1 0
,
 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 1 1 72 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 72 72
,
 6 67 0 0 0 0 6 6 0 0 0 0 0 0 0 0 72 1 0 0 1 1 71 72 0 0 25 25 72 0 0 0 25 24 72 0
,
 13 66 0 0 0 0 66 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 25 25 72 0 0 0 25 25 0 72

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,72,0,0,1,0,1,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,0,72,0,0,0,0,1,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,0,1,72,0,0,0,0,0,72],[6,6,0,0,0,0,67,6,0,0,0,0,0,0,0,1,25,25,0,0,0,1,25,24,0,0,72,71,72,72,0,0,1,72,0,0],[13,66,0,0,0,0,66,60,0,0,0,0,0,0,1,0,25,25,0,0,0,1,25,25,0,0,0,0,72,0,0,0,0,0,0,72] >;`

C3⋊S3.5Q16 in GAP, Magma, Sage, TeX

`C_3\rtimes S_3._5Q_{16}`
`% in TeX`

`G:=Group("C3:S3.5Q16");`
`// GroupNames label`

`G:=SmallGroup(288,432);`
`// by ID`

`G=gap.SmallGroup(288,432);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,675,346,80,9413,691,12550,2372]);`
`// 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=a^-1,d*a*d^-1=a*b^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;`
`// generators/relations`

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