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## G = C32⋊Q32order 288 = 25·32

### The semidirect product of C32 and Q32 acting via Q32/C4=D4

Aliases: C32⋊Q32, C4.3S3≀C2, (C3×C6).3D8, (C3×C12).7D4, C2.5(C32⋊D8), C322Q16.C2, C322C16.2C2, C324C8.3C22, SmallGroup(288,384)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊4C8 — C32⋊Q32
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2Q16 — C32⋊Q32
 Lower central C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊Q32
 Upper central C1 — C2 — C4

Generators and relations for C32⋊Q32
G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, cac-1=dad-1=b, cbc-1=a-1, dbd-1=a, dcd-1=c-1 >

Character table of C32⋊Q32

 class 1 2 3A 3B 4A 4B 4C 6A 6B 8A 8B 12A 12B 12C 12D 12E 12F 16A 16B 16C 16D size 1 1 4 4 2 24 24 4 4 18 18 8 8 24 24 24 24 18 18 18 18 ρ1 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 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 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 linear of order 2 ρ5 2 2 2 2 2 0 0 2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 2 2 -2 0 0 2 2 0 0 -2 -2 0 0 0 0 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ7 2 2 2 2 -2 0 0 2 2 0 0 -2 -2 0 0 0 0 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ8 2 -2 2 2 0 0 0 -2 -2 -√2 √2 0 0 0 0 0 0 -ζ167+ζ16 ζ167-ζ16 -ζ165+ζ163 ζ165-ζ163 symplectic lifted from Q32, Schur index 2 ρ9 2 -2 2 2 0 0 0 -2 -2 √2 -√2 0 0 0 0 0 0 ζ165-ζ163 -ζ165+ζ163 -ζ167+ζ16 ζ167-ζ16 symplectic lifted from Q32, Schur index 2 ρ10 2 -2 2 2 0 0 0 -2 -2 √2 -√2 0 0 0 0 0 0 -ζ165+ζ163 ζ165-ζ163 ζ167-ζ16 -ζ167+ζ16 symplectic lifted from Q32, Schur index 2 ρ11 2 -2 2 2 0 0 0 -2 -2 -√2 √2 0 0 0 0 0 0 ζ167-ζ16 -ζ167+ζ16 ζ165-ζ163 -ζ165+ζ163 symplectic lifted from Q32, Schur index 2 ρ12 4 4 1 -2 4 0 -2 1 -2 0 0 -2 1 0 1 0 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ13 4 4 1 -2 4 0 2 1 -2 0 0 -2 1 0 -1 0 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ14 4 4 -2 1 4 -2 0 -2 1 0 0 1 -2 1 0 1 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ15 4 4 -2 1 4 2 0 -2 1 0 0 1 -2 -1 0 -1 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ16 4 4 1 -2 -4 0 0 1 -2 0 0 2 -1 0 -√-3 0 √-3 0 0 0 0 complex lifted from C32⋊D8 ρ17 4 4 -2 1 -4 0 0 -2 1 0 0 -1 2 -√-3 0 √-3 0 0 0 0 0 complex lifted from C32⋊D8 ρ18 4 4 -2 1 -4 0 0 -2 1 0 0 -1 2 √-3 0 -√-3 0 0 0 0 0 complex lifted from C32⋊D8 ρ19 4 4 1 -2 -4 0 0 1 -2 0 0 2 -1 0 √-3 0 -√-3 0 0 0 0 complex lifted from C32⋊D8 ρ20 8 -8 -4 2 0 0 0 4 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ21 8 -8 2 -4 0 0 0 -2 4 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C32⋊Q32
On 96 points
Generators in S96
```(2 69 44)(4 46 71)(6 73 48)(8 34 75)(10 77 36)(12 38 79)(14 65 40)(16 42 67)(17 95 57)(19 59 81)(21 83 61)(23 63 85)(25 87 49)(27 51 89)(29 91 53)(31 55 93)
(1 68 43)(3 45 70)(5 72 47)(7 33 74)(9 76 35)(11 37 78)(13 80 39)(15 41 66)(18 58 96)(20 82 60)(22 62 84)(24 86 64)(26 50 88)(28 90 52)(30 54 92)(32 94 56)
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 59 9 51)(2 58 10 50)(3 57 11 49)(4 56 12 64)(5 55 13 63)(6 54 14 62)(7 53 15 61)(8 52 16 60)(17 37 25 45)(18 36 26 44)(19 35 27 43)(20 34 28 42)(21 33 29 41)(22 48 30 40)(23 47 31 39)(24 46 32 38)(65 84 73 92)(66 83 74 91)(67 82 75 90)(68 81 76 89)(69 96 77 88)(70 95 78 87)(71 94 79 86)(72 93 80 85)```

`G:=sub<Sym(96)| (2,69,44)(4,46,71)(6,73,48)(8,34,75)(10,77,36)(12,38,79)(14,65,40)(16,42,67)(17,95,57)(19,59,81)(21,83,61)(23,63,85)(25,87,49)(27,51,89)(29,91,53)(31,55,93), (1,68,43)(3,45,70)(5,72,47)(7,33,74)(9,76,35)(11,37,78)(13,80,39)(15,41,66)(18,58,96)(20,82,60)(22,62,84)(24,86,64)(26,50,88)(28,90,52)(30,54,92)(32,94,56), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,59,9,51)(2,58,10,50)(3,57,11,49)(4,56,12,64)(5,55,13,63)(6,54,14,62)(7,53,15,61)(8,52,16,60)(17,37,25,45)(18,36,26,44)(19,35,27,43)(20,34,28,42)(21,33,29,41)(22,48,30,40)(23,47,31,39)(24,46,32,38)(65,84,73,92)(66,83,74,91)(67,82,75,90)(68,81,76,89)(69,96,77,88)(70,95,78,87)(71,94,79,86)(72,93,80,85)>;`

`G:=Group( (2,69,44)(4,46,71)(6,73,48)(8,34,75)(10,77,36)(12,38,79)(14,65,40)(16,42,67)(17,95,57)(19,59,81)(21,83,61)(23,63,85)(25,87,49)(27,51,89)(29,91,53)(31,55,93), (1,68,43)(3,45,70)(5,72,47)(7,33,74)(9,76,35)(11,37,78)(13,80,39)(15,41,66)(18,58,96)(20,82,60)(22,62,84)(24,86,64)(26,50,88)(28,90,52)(30,54,92)(32,94,56), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,59,9,51)(2,58,10,50)(3,57,11,49)(4,56,12,64)(5,55,13,63)(6,54,14,62)(7,53,15,61)(8,52,16,60)(17,37,25,45)(18,36,26,44)(19,35,27,43)(20,34,28,42)(21,33,29,41)(22,48,30,40)(23,47,31,39)(24,46,32,38)(65,84,73,92)(66,83,74,91)(67,82,75,90)(68,81,76,89)(69,96,77,88)(70,95,78,87)(71,94,79,86)(72,93,80,85) );`

`G=PermutationGroup([(2,69,44),(4,46,71),(6,73,48),(8,34,75),(10,77,36),(12,38,79),(14,65,40),(16,42,67),(17,95,57),(19,59,81),(21,83,61),(23,63,85),(25,87,49),(27,51,89),(29,91,53),(31,55,93)], [(1,68,43),(3,45,70),(5,72,47),(7,33,74),(9,76,35),(11,37,78),(13,80,39),(15,41,66),(18,58,96),(20,82,60),(22,62,84),(24,86,64),(26,50,88),(28,90,52),(30,54,92),(32,94,56)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,59,9,51),(2,58,10,50),(3,57,11,49),(4,56,12,64),(5,55,13,63),(6,54,14,62),(7,53,15,61),(8,52,16,60),(17,37,25,45),(18,36,26,44),(19,35,27,43),(20,34,28,42),(21,33,29,41),(22,48,30,40),(23,47,31,39),(24,46,32,38),(65,84,73,92),(66,83,74,91),(67,82,75,90),(68,81,76,89),(69,96,77,88),(70,95,78,87),(71,94,79,86),(72,93,80,85)])`

Matrix representation of C32⋊Q32 in GL6(𝔽97)

 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 96 1 0 0 0 0 96 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 96 1 0 0 0 0 96 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 95 71 0 0 0 0 26 95 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 96 0 0 41 15 0 0 0 0 82 56 0 0
,
 40 40 0 0 0 0 40 57 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(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[95,26,0,0,0,0,71,95,0,0,0,0,0,0,0,0,41,82,0,0,0,0,15,56,0,0,1,1,0,0,0,0,0,96,0,0],[40,40,0,0,0,0,40,57,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C32⋊Q32 in GAP, Magma, Sage, TeX

`C_3^2\rtimes Q_{32}`
`% in TeX`

`G:=Group("C3^2:Q32");`
`// GroupNames label`

`G:=SmallGroup(288,384);`
`// by ID`

`G=gap.SmallGroup(288,384);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,85,120,254,135,142,675,346,80,2693,2028,691,797,2372]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,c*a*c^-1=d*a*d^-1=b,c*b*c^-1=a^-1,d*b*d^-1=a,d*c*d^-1=c^-1>;`
`// generators/relations`

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