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## G = C32⋊Q16⋊C2order 288 = 25·32

### 1st semidirect product of C32⋊Q16 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C32⋊Q16⋊C2
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊2SD16 — C32⋊Q16⋊C2
 Lower central C32 — C3×C6 — C3⋊Dic3 — C32⋊Q16⋊C2
 Upper central C1 — C2 — C4

Generators and relations for C32⋊Q16⋊C2
G = < a,b,c,d,e | a3=b3=c8=e2=1, d2=c4, ab=ba, cac-1=dad-1=b, eae=cbc-1=a-1, dbd-1=a, ebe=b-1, dcd-1=c-1, ece=c5, de=ed >

Subgroups: 496 in 99 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×4], C22 [×2], S3 [×4], C6 [×3], C8 [×2], C2×C4 [×3], D4 [×2], Q8 [×4], C32, Dic3 [×5], C12 [×5], D6 [×3], C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6 [×4], C4×S3 [×5], D12, C3⋊D4 [×2], C2×C12, C3×Q8, C8.C22, C3×Dic3 [×3], C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, S3×Q8, C322C8 [×2], C6.D6, D6⋊S3, C3⋊D12, C322Q8, C322Q8 [×2], C3×Dic6, S3×C12, C4×C3⋊S3, C322SD16 [×2], C32⋊Q16 [×2], C32⋊M4(2), Dic3.D6, D6.D6, C32⋊Q16⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C8.C22, S3≀C2, C2×S3≀C2, C32⋊Q16⋊C2

Character table of C32⋊Q16⋊C2

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E 12F 12G size 1 1 12 18 4 4 2 12 12 12 18 4 4 12 12 36 36 4 4 8 12 12 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 0 -2 2 2 2 0 0 0 -2 2 2 0 0 0 0 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 0 2 2 2 -2 0 0 0 -2 2 2 0 0 0 0 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 0 0 1 -2 -4 2 -2 0 0 1 -2 0 0 0 0 2 2 -1 0 0 -1 1 orthogonal lifted from C2×S3≀C2 ρ12 4 4 0 0 1 -2 4 -2 -2 0 0 1 -2 0 0 0 0 -2 -2 1 0 0 1 1 orthogonal lifted from S3≀C2 ρ13 4 4 -2 0 -2 1 4 0 0 -2 0 -2 1 1 1 0 0 1 1 -2 1 1 0 0 orthogonal lifted from S3≀C2 ρ14 4 4 2 0 -2 1 4 0 0 2 0 -2 1 -1 -1 0 0 1 1 -2 -1 -1 0 0 orthogonal lifted from S3≀C2 ρ15 4 4 0 0 1 -2 -4 -2 2 0 0 1 -2 0 0 0 0 2 2 -1 0 0 1 -1 orthogonal lifted from C2×S3≀C2 ρ16 4 4 -2 0 -2 1 -4 0 0 2 0 -2 1 1 1 0 0 -1 -1 2 -1 -1 0 0 orthogonal lifted from C2×S3≀C2 ρ17 4 4 0 0 1 -2 4 2 2 0 0 1 -2 0 0 0 0 -2 -2 1 0 0 -1 -1 orthogonal lifted from S3≀C2 ρ18 4 4 2 0 -2 1 -4 0 0 -2 0 -2 1 -1 -1 0 0 -1 -1 2 1 1 0 0 orthogonal lifted from C2×S3≀C2 ρ19 4 -4 0 0 4 4 0 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ20 4 -4 0 0 -2 1 0 0 0 0 0 2 -1 √-3 -√-3 0 0 -3i 3i 0 √3 -√3 0 0 complex faithful ρ21 4 -4 0 0 -2 1 0 0 0 0 0 2 -1 √-3 -√-3 0 0 3i -3i 0 -√3 √3 0 0 complex faithful ρ22 4 -4 0 0 -2 1 0 0 0 0 0 2 -1 -√-3 √-3 0 0 -3i 3i 0 -√3 √3 0 0 complex faithful ρ23 4 -4 0 0 -2 1 0 0 0 0 0 2 -1 -√-3 √-3 0 0 3i -3i 0 √3 -√3 0 0 complex faithful ρ24 8 -8 0 0 2 -4 0 0 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C32⋊Q16⋊C2
On 48 points
Generators in S48
```(2 10 46)(4 48 12)(6 14 42)(8 44 16)(17 32 33)(19 35 26)(21 28 37)(23 39 30)
(1 9 45)(3 47 11)(5 13 41)(7 43 15)(18 34 25)(20 27 36)(22 38 29)(24 31 40)
(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 21 5 17)(2 20 6 24)(3 19 7 23)(4 18 8 22)(9 28 13 32)(10 27 14 31)(11 26 15 30)(12 25 16 29)(33 45 37 41)(34 44 38 48)(35 43 39 47)(36 42 40 46)
(2 6)(4 8)(9 45)(10 42)(11 47)(12 44)(13 41)(14 46)(15 43)(16 48)(18 22)(20 24)(25 38)(26 35)(27 40)(28 37)(29 34)(30 39)(31 36)(32 33)```

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

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

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

Matrix representation of C32⋊Q16⋊C2 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 72 72 0 0 1 0
,
 72 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 0 0 7 14 0 0 7 66 43 13 0 0 60 30 0 0
,
 0 0 27 0 0 0 0 27 27 0 0 0 0 27 0 0
,
 1 0 0 0 72 72 0 0 0 0 1 0 0 0 72 72
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[0,0,43,60,0,0,13,30,7,7,0,0,14,66,0,0],[0,0,27,0,0,0,0,27,27,0,0,0,0,27,0,0],[1,72,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;`

C32⋊Q16⋊C2 in GAP, Magma, Sage, TeX

`C_3^2\rtimes Q_{16}\rtimes C_2`
`% in TeX`

`G:=Group("C3^2:Q16:C2");`
`// GroupNames label`

`G:=SmallGroup(288,874);`
`// by ID`

`G=gap.SmallGroup(288,874);`
`# by ID`

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

`G:=Group<a,b,c,d,e|a^3=b^3=c^8=e^2=1,d^2=c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=b,e*a*e=c*b*c^-1=a^-1,d*b*d^-1=a,e*b*e=b^-1,d*c*d^-1=c^-1,e*c*e=c^5,d*e=e*d>;`
`// generators/relations`

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