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## G = S33order 216 = 23·33

### Direct product of S3, S3 and S3

Aliases: S33, C33⋊C23, C3⋊S32D6, (C3×S3)⋊1D6, (S3×C32)⋊C22, C33⋊C2⋊C22, C324D63C2, C324(C22×S3), (S3×C3⋊S3)⋊C2, C31(C2×S32), (C3×S32)⋊3C2, (C3×C3⋊S3)⋊C22, Hol(C3×S3), SmallGroup(216,162)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S33
 Chief series C1 — C3 — C32 — C33 — S3×C32 — C3×S32 — S33
 Lower central C33 — S33
 Upper central C1

Generators and relations for S33
G = < a,b,c,d,e,f | a3=b2=c3=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 904 in 162 conjugacy classes, 38 normal (5 characteristic)
C1, C2, C3, C3, C22, S3, S3, C6, C23, C32, C32, D6, C2×C6, C3×S3, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C22×S3, C33, S32, S32, S3×C6, C2×C3⋊S3, S3×C32, C3×C3⋊S3, C33⋊C2, C2×S32, C3×S32, S3×C3⋊S3, C324D6, S33
Quotients: C1, C2, C22, S3, C23, D6, C22×S3, S32, C2×S32, S33

Character table of S33

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L size 1 3 3 3 9 9 9 27 2 2 2 4 4 4 8 6 6 6 6 6 6 12 12 12 18 18 18 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 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 -2 2 0 0 0 -2 0 -1 2 2 -1 -1 2 -1 2 -2 0 0 1 -1 0 -1 1 1 0 0 orthogonal lifted from D6 ρ10 2 0 -2 -2 2 0 0 0 2 2 -1 2 -1 -1 -1 1 0 -2 1 0 -2 1 1 0 0 -1 0 orthogonal lifted from D6 ρ11 2 2 0 2 0 2 0 0 2 -1 2 -1 2 -1 -1 0 -1 -1 2 2 0 -1 0 -1 0 0 -1 orthogonal lifted from S3 ρ12 2 0 2 2 2 0 0 0 2 2 -1 2 -1 -1 -1 -1 0 2 -1 0 2 -1 -1 0 0 -1 0 orthogonal lifted from S3 ρ13 2 -2 0 -2 0 2 0 0 2 -1 2 -1 2 -1 -1 0 1 1 -2 -2 0 1 0 1 0 0 -1 orthogonal lifted from D6 ρ14 2 -2 -2 0 0 0 2 0 -1 2 2 -1 -1 2 -1 -2 -2 0 0 1 1 0 1 1 -1 0 0 orthogonal lifted from D6 ρ15 2 0 -2 2 -2 0 0 0 2 2 -1 2 -1 -1 -1 1 0 2 -1 0 -2 -1 1 0 0 1 0 orthogonal lifted from D6 ρ16 2 2 -2 0 0 0 -2 0 -1 2 2 -1 -1 2 -1 -2 2 0 0 -1 1 0 1 -1 1 0 0 orthogonal lifted from D6 ρ17 2 -2 0 2 0 -2 0 0 2 -1 2 -1 2 -1 -1 0 1 -1 2 -2 0 -1 0 1 0 0 1 orthogonal lifted from D6 ρ18 2 2 2 0 0 0 2 0 -1 2 2 -1 -1 2 -1 2 2 0 0 -1 -1 0 -1 -1 -1 0 0 orthogonal lifted from S3 ρ19 2 2 0 -2 0 -2 0 0 2 -1 2 -1 2 -1 -1 0 -1 1 -2 2 0 1 0 -1 0 0 1 orthogonal lifted from D6 ρ20 2 0 2 -2 -2 0 0 0 2 2 -1 2 -1 -1 -1 -1 0 -2 1 0 2 1 -1 0 0 1 0 orthogonal lifted from D6 ρ21 4 0 4 0 0 0 0 0 -2 4 -2 -2 1 -2 1 -2 0 0 0 0 -2 0 1 0 0 0 0 orthogonal lifted from S32 ρ22 4 0 0 -4 0 0 0 0 4 -2 -2 -2 -2 1 1 0 0 2 2 0 0 -1 0 0 0 0 0 orthogonal lifted from C2×S32 ρ23 4 0 -4 0 0 0 0 0 -2 4 -2 -2 1 -2 1 2 0 0 0 0 2 0 -1 0 0 0 0 orthogonal lifted from C2×S32 ρ24 4 0 0 4 0 0 0 0 4 -2 -2 -2 -2 1 1 0 0 -2 -2 0 0 1 0 0 0 0 0 orthogonal lifted from S32 ρ25 4 -4 0 0 0 0 0 0 -2 -2 4 1 -2 -2 1 0 2 0 0 2 0 0 0 -1 0 0 0 orthogonal lifted from C2×S32 ρ26 4 4 0 0 0 0 0 0 -2 -2 4 1 -2 -2 1 0 -2 0 0 -2 0 0 0 1 0 0 0 orthogonal lifted from S32 ρ27 8 0 0 0 0 0 0 0 -4 -4 -4 2 2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of S33
On 12 points - transitive group 12T117
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 9 8)(10 11 12)
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)
(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)(7,9,8)(10,11,12), (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,2,3)(4,6,5)(7,9,8)(10,11,12), (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)(7,9,8)(10,11,12), (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,2,3)(4,6,5)(7,9,8)(10,11,12), (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),(7,9,8),(10,11,12)], [(1,10),(2,11),(3,12),(4,8),(5,9),(6,7)], [(1,2,3),(4,6,5),(7,9,8),(10,11,12)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12)]])`

`G:=TransitiveGroup(12,117);`

On 18 points - transitive group 18T96
Generators in S18
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 10)(2 12)(3 11)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)
(1 10)(2 11)(3 12)(4 15)(5 13)(6 14)(7 18)(8 16)(9 17)
(1 5 8)(2 6 9)(3 4 7)(10 13 16)(11 14 17)(12 15 18)
(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)```

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

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

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

`G:=TransitiveGroup(18,96);`

On 24 points - transitive group 24T557
Generators in S24
```(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 4)(2 6)(3 5)(7 11)(8 10)(9 12)(13 17)(14 16)(15 18)(19 23)(20 22)(21 24)
(1 3 2)(4 5 6)(7 9 8)(10 11 12)(13 14 15)(16 18 17)(19 20 21)(22 24 23)
(1 14)(2 15)(3 13)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 2 3)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 18 17)(19 21 20)(22 23 24)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)```

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

`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), (1,4)(2,6)(3,5)(7,11)(8,10)(9,12)(13,17)(14,16)(15,18)(19,23)(20,22)(21,24), (1,3,2)(4,5,6)(7,9,8)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,24,23), (1,14)(2,15)(3,13)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,23,24), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24) );`

`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)], [(1,4),(2,6),(3,5),(7,11),(8,10),(9,12),(13,17),(14,16),(15,18),(19,23),(20,22),(21,24)], [(1,3,2),(4,5,6),(7,9,8),(10,11,12),(13,14,15),(16,18,17),(19,20,21),(22,24,23)], [(1,14),(2,15),(3,13),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,2,3),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,18,17),(19,21,20),(22,23,24)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)]])`

`G:=TransitiveGroup(24,557);`

On 27 points - transitive group 27T86
Generators in S27
```(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)
(2 3)(5 6)(8 9)(10 12)(14 15)(17 18)(19 21)(23 24)(26 27)
(1 13 25)(2 14 26)(3 15 27)(4 20 7)(5 21 8)(6 19 9)(10 18 24)(11 16 22)(12 17 23)
(4 20)(5 21)(6 19)(10 24)(11 22)(12 23)(13 25)(14 26)(15 27)
(1 7 16)(2 8 17)(3 9 18)(4 22 13)(5 23 14)(6 24 15)(10 27 19)(11 25 20)(12 26 21)
(4 22)(5 23)(6 24)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)```

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

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

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

`G:=TransitiveGroup(27,86);`

Polynomial with Galois group S33 over ℚ
actionf(x)Disc(f)
12T117x12-4x9+2x6+4x3-2226·318

Matrix representation of S33 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 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 0 1 0 0 0 0 1 0 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 -1 1 0 0 0 0 -1 0
,
 -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 0 -1 0 0 0 0 -1 0
,
 -1 1 0 0 0 0 -1 0 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
,
 0 -1 0 0 0 0 -1 0 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

`G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,-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,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

S33 in GAP, Magma, Sage, TeX

`S_3^3`
`% in TeX`

`G:=Group("S3^3");`
`// GroupNames label`

`G:=SmallGroup(216,162);`
`// by ID`

`G=gap.SmallGroup(216,162);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,111,730,5189]);`
`// Polycyclic`

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

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