direct product, metabelian, supersoluble, monomial, A-group, rational
Aliases: S32, Spin+4(𝔽2), Ω+4(𝔽2), PΩ+4(𝔽2), C3⋊1D6, C32⋊C22, C3⋊S3⋊C2, (C3×S3)⋊C2, Hol(S3), SmallGroup(36,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — S32 |
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 |
►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 C32⋊4D6 D15⋊S3 D21⋊S3 D33⋊S3 D39⋊S3 ...
S32 is a maximal quotient of
C6.D6 C32⋊2Q8
C32⋊D2p: D6⋊S3 C3⋊D12 C32⋊D6 C32⋊4D6 D15⋊S3 D21⋊S3 D33⋊S3 D39⋊S3 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 6T9 | x6-x3+2 | -22·36·73 |
| 9T8 | x9-16x7-7x6+48x5+20x4-37x3-16x2+4x+1 | 26·373·1013·9472 |
| 12T16 | x12-4x11-16x10+92x9-15x8-536x7+878x6+52x5-1109x4+720x3+38x2-104x+4 | 230·56·194·594·612 |
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
Subgroup lattice of S32 in TeX
Character table of S32 in TeX