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## G = C3⋊Q32order 96 = 25·3

### The semidirect product of C3 and Q32 acting via Q32/Q16=C2

Aliases: C32Q32, C8.7D6, Q16.S3, C12.6D4, C6.11D8, C24.5C22, Dic12.2C2, C3⋊C16.C2, C2.7(D4⋊S3), C4.4(C3⋊D4), (C3×Q16).1C2, SmallGroup(96,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3⋊Q32
 Chief series C1 — C3 — C6 — C12 — C24 — Dic12 — C3⋊Q32
 Lower central C3 — C6 — C12 — C24 — C3⋊Q32
 Upper central C1 — C2 — C4 — C8 — Q16

Generators and relations for C3⋊Q32
G = < a,b,c | a3=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=b-1 >

Character table of C3⋊Q32

 class 1 2 3 4A 4B 4C 6 8A 8B 12A 12B 12C 16A 16B 16C 16D 24A 24B size 1 1 2 2 8 24 2 2 2 4 8 8 6 6 6 6 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 -1 2 -2 0 -1 2 2 -1 1 1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ6 2 2 -1 2 2 0 -1 2 2 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 0 0 2 -2 -2 2 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ8 2 2 2 -2 0 0 2 0 0 -2 0 0 -√2 √2 -√2 √2 0 0 orthogonal lifted from D8 ρ9 2 2 2 -2 0 0 2 0 0 -2 0 0 √2 -√2 √2 -√2 0 0 orthogonal lifted from D8 ρ10 2 -2 2 0 0 0 -2 -√2 √2 0 0 0 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 √2 -√2 symplectic lifted from Q32, Schur index 2 ρ11 2 -2 2 0 0 0 -2 √2 -√2 0 0 0 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -√2 √2 symplectic lifted from Q32, Schur index 2 ρ12 2 -2 2 0 0 0 -2 -√2 √2 0 0 0 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 √2 -√2 symplectic lifted from Q32, Schur index 2 ρ13 2 -2 2 0 0 0 -2 √2 -√2 0 0 0 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 -√2 √2 symplectic lifted from Q32, Schur index 2 ρ14 2 2 -1 2 0 0 -1 -2 -2 -1 -√-3 √-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ15 2 2 -1 2 0 0 -1 -2 -2 -1 √-3 -√-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ16 4 4 -2 -4 0 0 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ17 4 -4 -2 0 0 0 2 2√2 -2√2 0 0 0 0 0 0 0 √2 -√2 symplectic faithful, Schur index 2 ρ18 4 -4 -2 0 0 0 2 -2√2 2√2 0 0 0 0 0 0 0 -√2 √2 symplectic faithful, Schur index 2

Smallest permutation representation of C3⋊Q32
Regular action on 96 points
Generators in S96
```(1 30 49)(2 50 31)(3 32 51)(4 52 17)(5 18 53)(6 54 19)(7 20 55)(8 56 21)(9 22 57)(10 58 23)(11 24 59)(12 60 25)(13 26 61)(14 62 27)(15 28 63)(16 64 29)(33 84 73)(34 74 85)(35 86 75)(36 76 87)(37 88 77)(38 78 89)(39 90 79)(40 80 91)(41 92 65)(42 66 93)(43 94 67)(44 68 95)(45 96 69)(46 70 81)(47 82 71)(48 72 83)
(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 36 9 44)(2 35 10 43)(3 34 11 42)(4 33 12 41)(5 48 13 40)(6 47 14 39)(7 46 15 38)(8 45 16 37)(17 73 25 65)(18 72 26 80)(19 71 27 79)(20 70 28 78)(21 69 29 77)(22 68 30 76)(23 67 31 75)(24 66 32 74)(49 87 57 95)(50 86 58 94)(51 85 59 93)(52 84 60 92)(53 83 61 91)(54 82 62 90)(55 81 63 89)(56 96 64 88)```

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

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

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

C3⋊Q32 is a maximal subgroup of
SD32⋊S3  D6.2D8  S3×Q32  Q32⋊S3  C24.27C23  Q16.D6  D8.9D6  C9⋊Q32  C322Q32  C323Q32  C327Q32  C15⋊Q32  C3⋊Dic40  C157Q32
C3⋊Q32 is a maximal quotient of
C6.6D16  C6.Q32  C6.5Q32  C9⋊Q32  C322Q32  C323Q32  C327Q32  C15⋊Q32  C3⋊Dic40  C157Q32

Matrix representation of C3⋊Q32 in GL4(𝔽7) generated by

 0 0 5 2 5 3 0 3 2 5 0 3 5 5 1 2
,
 4 2 3 2 1 0 0 5 1 6 3 4 0 0 1 0
,
 3 0 1 2 1 0 6 6 2 2 5 3 1 6 3 6
`G:=sub<GL(4,GF(7))| [0,5,2,5,0,3,5,5,5,0,0,1,2,3,3,2],[4,1,1,0,2,0,6,0,3,0,3,1,2,5,4,0],[3,1,2,1,0,0,2,6,1,6,5,3,2,6,3,6] >;`

C3⋊Q32 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(96,36);`
`// by ID`

`G=gap.SmallGroup(96,36);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,73,103,218,116,122,579,297,69,2309]);`
`// Polycyclic`

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

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