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## G = S32⋊D5order 360 = 23·32·5

### The semidirect product of S32 and D5 acting via D5/C5=C2

Aliases: S32⋊D5, C52S3≀C2, (C3×C15)⋊1D4, C32⋊(C5⋊D4), D15⋊S31C2, C3⋊S3.1D10, C32⋊Dic52C2, (C5×S32)⋊3C2, (C5×C3⋊S3).2C22, SmallGroup(360,133)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×C3⋊S3 — S32⋊D5
 Chief series C1 — C5 — C3×C15 — C5×C3⋊S3 — D15⋊S3 — S32⋊D5
 Lower central C3×C15 — C5×C3⋊S3 — S32⋊D5
 Upper central C1

Generators and relations for S32⋊D5
G = < a,b,c,d,e | a5=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

6C2
9C2
30C2
2C3
2C3
9C22
45C22
45C4
2S3
6S3
6C6
6S3
10S3
30C6
6D5
6C10
9C10
2C15
2C15
45D4
6D6
30D6
10C3×S3
9D10
9Dic5
2D15
6C30
5S32

Character table of S32⋊D5

 class 1 2A 2B 2C 3A 3B 4 5A 5B 6A 6B 10A 10B 10C 10D 10E 10F 15A 15B 15C 15D 15E 15F 30A 30B 30C 30D size 1 6 9 30 4 4 90 2 2 12 60 6 6 6 6 18 18 4 4 4 4 8 8 12 12 12 12 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 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 -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 -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 1 1 1 linear of order 2 ρ5 2 0 -2 0 2 2 0 2 2 0 0 0 0 0 0 -2 -2 2 2 2 2 2 2 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 2 0 2 2 0 -1-√5/2 -1+√5/2 -2 0 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ7 2 2 2 0 2 2 0 -1+√5/2 -1-√5/2 2 0 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ8 2 2 2 0 2 2 0 -1-√5/2 -1+√5/2 2 0 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 -2 2 0 2 2 0 -1+√5/2 -1-√5/2 -2 0 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ10 2 0 -2 0 2 2 0 -1+√5/2 -1-√5/2 0 0 ζ53-ζ52 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 complex lifted from C5⋊D4 ρ11 2 0 -2 0 2 2 0 -1+√5/2 -1-√5/2 0 0 -ζ53+ζ52 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 complex lifted from C5⋊D4 ρ12 2 0 -2 0 2 2 0 -1-√5/2 -1+√5/2 0 0 -ζ54+ζ5 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 complex lifted from C5⋊D4 ρ13 2 0 -2 0 2 2 0 -1-√5/2 -1+√5/2 0 0 ζ54-ζ5 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 1+√5/2 1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 complex lifted from C5⋊D4 ρ14 4 0 0 -2 1 -2 0 4 4 0 1 0 0 0 0 0 0 -2 -2 -2 -2 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ15 4 -2 0 0 -2 1 0 4 4 1 0 -2 -2 -2 -2 0 0 1 1 1 1 -2 -2 1 1 1 1 orthogonal lifted from S3≀C2 ρ16 4 0 0 2 1 -2 0 4 4 0 -1 0 0 0 0 0 0 -2 -2 -2 -2 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 2 0 0 -2 1 0 4 4 -1 0 2 2 2 2 0 0 1 1 1 1 -2 -2 -1 -1 -1 -1 orthogonal lifted from S3≀C2 ρ18 4 -2 0 0 -2 1 0 -1+√5 -1-√5 1 0 -2ζ53 -2ζ52 -2ζ54 -2ζ5 0 0 2ζ53-ζ52 2ζ54-ζ5 -ζ53+2ζ52 -ζ54+2ζ5 1+√5/2 1-√5/2 ζ54 ζ5 ζ53 ζ52 complex faithful ρ19 4 2 0 0 -2 1 0 -1-√5 -1+√5 -1 0 2ζ54 2ζ5 2ζ52 2ζ53 0 0 2ζ54-ζ5 -ζ53+2ζ52 -ζ54+2ζ5 2ζ53-ζ52 1-√5/2 1+√5/2 -ζ52 -ζ53 -ζ54 -ζ5 complex faithful ρ20 4 2 0 0 -2 1 0 -1-√5 -1+√5 -1 0 2ζ5 2ζ54 2ζ53 2ζ52 0 0 -ζ54+2ζ5 2ζ53-ζ52 2ζ54-ζ5 -ζ53+2ζ52 1-√5/2 1+√5/2 -ζ53 -ζ52 -ζ5 -ζ54 complex faithful ρ21 4 2 0 0 -2 1 0 -1+√5 -1-√5 -1 0 2ζ53 2ζ52 2ζ54 2ζ5 0 0 2ζ53-ζ52 2ζ54-ζ5 -ζ53+2ζ52 -ζ54+2ζ5 1+√5/2 1-√5/2 -ζ54 -ζ5 -ζ53 -ζ52 complex faithful ρ22 4 -2 0 0 -2 1 0 -1-√5 -1+√5 1 0 -2ζ54 -2ζ5 -2ζ52 -2ζ53 0 0 2ζ54-ζ5 -ζ53+2ζ52 -ζ54+2ζ5 2ζ53-ζ52 1-√5/2 1+√5/2 ζ52 ζ53 ζ54 ζ5 complex faithful ρ23 4 -2 0 0 -2 1 0 -1-√5 -1+√5 1 0 -2ζ5 -2ζ54 -2ζ53 -2ζ52 0 0 -ζ54+2ζ5 2ζ53-ζ52 2ζ54-ζ5 -ζ53+2ζ52 1-√5/2 1+√5/2 ζ53 ζ52 ζ5 ζ54 complex faithful ρ24 4 -2 0 0 -2 1 0 -1+√5 -1-√5 1 0 -2ζ52 -2ζ53 -2ζ5 -2ζ54 0 0 -ζ53+2ζ52 -ζ54+2ζ5 2ζ53-ζ52 2ζ54-ζ5 1+√5/2 1-√5/2 ζ5 ζ54 ζ52 ζ53 complex faithful ρ25 4 2 0 0 -2 1 0 -1+√5 -1-√5 -1 0 2ζ52 2ζ53 2ζ5 2ζ54 0 0 -ζ53+2ζ52 -ζ54+2ζ5 2ζ53-ζ52 2ζ54-ζ5 1+√5/2 1-√5/2 -ζ5 -ζ54 -ζ52 -ζ53 complex faithful ρ26 8 0 0 0 2 -4 0 -2+2√5 -2-2√5 0 0 0 0 0 0 0 0 1+√5 1-√5 1+√5 1-√5 -1-√5/2 -1+√5/2 0 0 0 0 orthogonal faithful ρ27 8 0 0 0 2 -4 0 -2-2√5 -2+2√5 0 0 0 0 0 0 0 0 1-√5 1+√5 1-√5 1+√5 -1+√5/2 -1-√5/2 0 0 0 0 orthogonal faithful

Permutation representations of S32⋊D5
On 30 points - transitive group 30T96
Generators in S30
```(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)
(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)
(1 19)(2 18)(3 17)(4 16)(5 20)(6 22 11 27)(7 21 12 26)(8 25 13 30)(9 24 14 29)(10 23 15 28)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 27)(22 26)(23 30)(24 29)(25 28)```

`G:=sub<Sym(30)| (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), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (1,19)(2,18)(3,17)(4,16)(5,20)(6,22,11,27)(7,21,12,26)(8,25,13,30)(9,24,14,29)(10,23,15,28), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,27)(22,26)(23,30)(24,29)(25,28)>;`

`G:=Group( (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), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (1,19)(2,18)(3,17)(4,16)(5,20)(6,22,11,27)(7,21,12,26)(8,25,13,30)(9,24,14,29)(10,23,15,28), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,27)(22,26)(23,30)(24,29)(25,28) );`

`G=PermutationGroup([[(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)], [(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13)], [(1,19),(2,18),(3,17),(4,16),(5,20),(6,22,11,27),(7,21,12,26),(8,25,13,30),(9,24,14,29),(10,23,15,28)], [(2,5),(3,4),(6,7),(8,10),(11,12),(13,15),(16,17),(18,20),(21,27),(22,26),(23,30),(24,29),(25,28)]])`

`G:=TransitiveGroup(30,96);`

On 30 points - transitive group 30T100
Generators in S30
```(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)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 16)(2 20)(3 19)(4 18)(5 17)(6 29 11 24)(7 28 12 23)(8 27 13 22)(9 26 14 21)(10 30 15 25)
(1 16)(2 20)(3 19)(4 18)(5 17)(6 24)(7 23)(8 22)(9 21)(10 25)(11 29)(12 28)(13 27)(14 26)(15 30)```

`G:=sub<Sym(30)| (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), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,29,11,24)(7,28,12,23)(8,27,13,22)(9,26,14,21)(10,30,15,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(9,21)(10,25)(11,29)(12,28)(13,27)(14,26)(15,30)>;`

`G:=Group( (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), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,29,11,24)(7,28,12,23)(8,27,13,22)(9,26,14,21)(10,30,15,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(9,21)(10,25)(11,29)(12,28)(13,27)(14,26)(15,30) );`

`G=PermutationGroup([[(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)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,29,11,24),(7,28,12,23),(8,27,13,22),(9,26,14,21),(10,30,15,25)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,24),(7,23),(8,22),(9,21),(10,25),(11,29),(12,28),(13,27),(14,26),(15,30)]])`

`G:=TransitiveGroup(30,100);`

Matrix representation of S32⋊D5 in GL6(𝔽61)

 34 0 0 0 0 0 0 9 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 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 59 15 0 0 0 0 12 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 59 15 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 60 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 12 1 0 0 60 0 0 0 0 0 0 60 0 0
,
 0 60 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 12 1

`G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,12,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,12,0,0,0,0,15,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,60,12,0,0,0,0,0,1,0,0],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,12,0,0,0,0,0,1] >;`

S32⋊D5 in GAP, Magma, Sage, TeX

`S_3^2\rtimes D_5`
`% in TeX`

`G:=Group("S3^2:D5");`
`// GroupNames label`

`G:=SmallGroup(360,133);`
`// by ID`

`G=gap.SmallGroup(360,133);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,3,-5,73,579,201,111,244,376,10373]);`
`// Polycyclic`

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

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