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G = S32order 36 = 22·32

Direct product of S3 and S3

direct product, metabelian, supersoluble, monomial, A-group, rational

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

C1C32 — S32
C1C3C32C3×S3 — S32
C32 — S32
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 >

3C2
3C2
9C2
2C3
9C22
3S3
3C6
3C6
3S3
6S3
3D6
3D6

Character table of S32

 class 12A2B2C3A3B3C6A6B
 size 133922466
ρ1111111111    trivial
ρ21-11-11111-1    linear of order 2
ρ311-1-1111-11    linear of order 2
ρ41-1-11111-1-1    linear of order 2
ρ522002-1-10-1    orthogonal lifted from S3
ρ62-2002-1-101    orthogonal lifted from D6
ρ72020-12-1-10    orthogonal lifted from S3
ρ820-20-12-110    orthogonal lifted from D6
ρ94000-2-2100    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);

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

0100
-1-100
00-1-1
0010
,
0010
0001
1000
0100
,
-1-100
1000
00-1-1
0010
,
0010
00-1-1
1000
-1-100
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

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