Copied to
clipboard

## G = S32order 36 = 22·32

### Direct product of S3 and S3

Aliases: S32, Spin+4(𝔽2), Ω+4(𝔽2), PΩ+4(𝔽2), C31D6, C32⋊C22, C3⋊S3⋊C2, (C3×S3)⋊C2, Hol(S3), SmallGroup(36,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32
 Chief series C1 — C3 — C32 — C3×S3 — S32
 Lower central C32 — S32
 Upper central C1

Generators and relations for S32
G = < a,b,c,d | a3=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of S32

 class 1 2A 2B 2C 3A 3B 3C 6A 6B size 1 3 3 9 2 2 4 6 6 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 1 1 -1 linear of order 2 ρ3 1 1 -1 -1 1 1 1 -1 1 linear of order 2 ρ4 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ5 2 2 0 0 2 -1 -1 0 -1 orthogonal lifted from S3 ρ6 2 -2 0 0 2 -1 -1 0 1 orthogonal lifted from D6 ρ7 2 0 2 0 -1 2 -1 -1 0 orthogonal lifted from S3 ρ8 2 0 -2 0 -1 2 -1 1 0 orthogonal lifted from D6 ρ9 4 0 0 0 -2 -2 1 0 0 orthogonal faithful

Permutation representations of S32
On 6 points - transitive group 6T9
Generators in S6
```(1 2 3)(4 5 6)
(1 4)(2 6)(3 5)
(1 3 2)(4 5 6)
(1 4)(2 5)(3 6)```

`G:=sub<Sym(6)| (1,2,3)(4,5,6), (1,4)(2,6)(3,5), (1,3,2)(4,5,6), (1,4)(2,5)(3,6)>;`

`G:=Group( (1,2,3)(4,5,6), (1,4)(2,6)(3,5), (1,3,2)(4,5,6), (1,4)(2,5)(3,6) );`

`G=PermutationGroup([[(1,2,3),(4,5,6)], [(1,4),(2,6),(3,5)], [(1,3,2),(4,5,6)], [(1,4),(2,5),(3,6)]])`

`G:=TransitiveGroup(6,9);`

On 9 points - transitive group 9T8
Generators in S9
```(1 2 3)(4 5 6)(7 8 9)
(2 3)(4 6)(7 9)
(1 8 5)(2 9 6)(3 7 4)
(4 7)(5 8)(6 9)```

`G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (2,3)(4,6)(7,9), (1,8,5)(2,9,6)(3,7,4), (4,7)(5,8)(6,9)>;`

`G:=Group( (1,2,3)(4,5,6)(7,8,9), (2,3)(4,6)(7,9), (1,8,5)(2,9,6)(3,7,4), (4,7)(5,8)(6,9) );`

`G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9)], [(2,3),(4,6),(7,9)], [(1,8,5),(2,9,6),(3,7,4)], [(4,7),(5,8),(6,9)]])`

`G:=TransitiveGroup(9,8);`

On 12 points - transitive group 12T16
Generators in S12
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 4)(2 6)(3 5)(7 11)(8 10)(9 12)
(1 3 2)(4 5 6)(7 8 9)(10 12 11)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)```

`G:=sub<Sym(12)| (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,4)(2,6)(3,5)(7,11)(8,10)(9,12), (1,3,2)(4,5,6)(7,8,9)(10,12,11), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)>;`

`G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,4)(2,6)(3,5)(7,11)(8,10)(9,12), (1,3,2)(4,5,6)(7,8,9)(10,12,11), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12) );`

`G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,4),(2,6),(3,5),(7,11),(8,10),(9,12)], [(1,3,2),(4,5,6),(7,8,9),(10,12,11)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12)]])`

`G:=TransitiveGroup(12,16);`

On 18 points - transitive group 18T9
Generators in S18
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 16)(2 18)(3 17)(4 15)(5 14)(6 13)(7 11)(8 10)(9 12)
(1 10 13)(2 11 14)(3 12 15)(4 17 9)(5 18 7)(6 16 8)
(1 16)(2 17)(3 18)(4 11)(5 12)(6 10)(7 15)(8 13)(9 14)```

`G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,16)(2,18)(3,17)(4,15)(5,14)(6,13)(7,11)(8,10)(9,12), (1,10,13)(2,11,14)(3,12,15)(4,17,9)(5,18,7)(6,16,8), (1,16)(2,17)(3,18)(4,11)(5,12)(6,10)(7,15)(8,13)(9,14)>;`

`G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,16)(2,18)(3,17)(4,15)(5,14)(6,13)(7,11)(8,10)(9,12), (1,10,13)(2,11,14)(3,12,15)(4,17,9)(5,18,7)(6,16,8), (1,16)(2,17)(3,18)(4,11)(5,12)(6,10)(7,15)(8,13)(9,14) );`

`G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,16),(2,18),(3,17),(4,15),(5,14),(6,13),(7,11),(8,10),(9,12)], [(1,10,13),(2,11,14),(3,12,15),(4,17,9),(5,18,7),(6,16,8)], [(1,16),(2,17),(3,18),(4,11),(5,12),(6,10),(7,15),(8,13),(9,14)]])`

`G:=TransitiveGroup(18,9);`

On 18 points - transitive group 18T11
Generators in S18
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 16)(2 18)(3 17)(4 15)(5 14)(6 13)(7 11)(8 10)(9 12)
(1 10 13)(2 11 14)(3 12 15)(4 17 9)(5 18 7)(6 16 8)
(4 9)(5 7)(6 8)(10 13)(11 14)(12 15)```

`G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,16)(2,18)(3,17)(4,15)(5,14)(6,13)(7,11)(8,10)(9,12), (1,10,13)(2,11,14)(3,12,15)(4,17,9)(5,18,7)(6,16,8), (4,9)(5,7)(6,8)(10,13)(11,14)(12,15)>;`

`G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,16)(2,18)(3,17)(4,15)(5,14)(6,13)(7,11)(8,10)(9,12), (1,10,13)(2,11,14)(3,12,15)(4,17,9)(5,18,7)(6,16,8), (4,9)(5,7)(6,8)(10,13)(11,14)(12,15) );`

`G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,16),(2,18),(3,17),(4,15),(5,14),(6,13),(7,11),(8,10),(9,12)], [(1,10,13),(2,11,14),(3,12,15),(4,17,9),(5,18,7),(6,16,8)], [(4,9),(5,7),(6,8),(10,13),(11,14),(12,15)]])`

`G:=TransitiveGroup(18,11);`

S32 is a maximal subgroup of
ΓL2(𝔽4)
C32⋊D2p: S3≀C2  C32⋊D6  C324D6  D15⋊S3  D21⋊S3  D33⋊S3  D39⋊S3 ...
S32 is a maximal quotient of
C6.D6  C322Q8
C32⋊D2p: D6⋊S3  C3⋊D12  C32⋊D6  C324D6  D15⋊S3  D21⋊S3  D33⋊S3  D39⋊S3 ...

Polynomial with Galois group S32 over ℚ
actionf(x)Disc(f)
6T9x6-x3+2-22·36·73
9T8x9-16x7-7x6+48x5+20x4-37x3-16x2+4x+126·373·1013·9472
12T16x12-4x11-16x10+92x9-15x8-536x7+878x6+52x5-1109x4+720x3+38x2-104x+4230·56·194·594·612

Matrix representation of S32 in GL4(ℤ) generated by

 0 1 0 0 -1 -1 0 0 0 0 -1 -1 0 0 1 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
,
 -1 -1 0 0 1 0 0 0 0 0 -1 -1 0 0 1 0
,
 0 0 1 0 0 0 -1 -1 1 0 0 0 -1 -1 0 0
`G:=sub<GL(4,Integers())| [0,-1,0,0,1,-1,0,0,0,0,-1,1,0,0,-1,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[-1,1,0,0,-1,0,0,0,0,0,-1,1,0,0,-1,0],[0,0,1,-1,0,0,0,-1,1,-1,0,0,0,-1,0,0] >;`

S32 in GAP, Magma, Sage, TeX

`S_3^2`
`% in TeX`

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

`G:=SmallGroup(36,10);`
`// by ID`

`G=gap.SmallGroup(36,10);`
`# by ID`

`G:=PCGroup([4,-2,-2,-3,-3,54,387]);`
`// Polycyclic`

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

Export

׿
×
𝔽