non-abelian, soluble, monomial, rational
Aliases: S3≀C2, SO+4(𝔽2), O+4(𝔽2), PSO+4(𝔽2), PO+4(𝔽2), CO+4(𝔽2), CSO+4(𝔽2), PCO+4(𝔽2), PCSO+4(𝔽2), C32⋊D4, S32⋊C2, C32⋊C4⋊C2, C3⋊S3.1C22, SmallGroup(72,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3≀C2
G = < a,b,c,d | a3=b3=c4=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >
Character table of S3≀C2
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | |
| size | 1 | 6 | 6 | 9 | 4 | 4 | 18 | 12 | 12 | |
| 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 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 4 | 0 | -2 | 0 | -2 | 1 | 0 | 1 | 0 | orthogonal faithful |
| ρ7 | 4 | -2 | 0 | 0 | 1 | -2 | 0 | 0 | 1 | orthogonal faithful |
| ρ8 | 4 | 0 | 2 | 0 | -2 | 1 | 0 | -1 | 0 | orthogonal faithful |
| ρ9 | 4 | 2 | 0 | 0 | 1 | -2 | 0 | 0 | -1 | orthogonal faithful |
►On 6 points - transitive group 6T13
Generators in S6
(1 4 6)(2 3 5)
(1 6 4)(2 3 5)
(1 2)(3 4 5 6)
(1 2)(3 6)(4 5)
G:=sub<Sym(6)| (1,4,6)(2,3,5), (1,6,4)(2,3,5), (1,2)(3,4,5,6), (1,2)(3,6)(4,5)>;
G:=Group( (1,4,6)(2,3,5), (1,6,4)(2,3,5), (1,2)(3,4,5,6), (1,2)(3,6)(4,5) );
G=PermutationGroup([[(1,4,6),(2,3,5)], [(1,6,4),(2,3,5)], [(1,2),(3,4,5,6)], [(1,2),(3,6),(4,5)]])
G:=TransitiveGroup(6,13);
►On 9 points: primitive - transitive group 9T16
Generators in S9
(1 5 3)(2 8 9)(4 7 6)
(1 4 2)(3 6 9)(5 7 8)
(2 3 4 5)(6 7 8 9)
(3 5)(6 7)(8 9)
G:=sub<Sym(9)| (1,5,3)(2,8,9)(4,7,6), (1,4,2)(3,6,9)(5,7,8), (2,3,4,5)(6,7,8,9), (3,5)(6,7)(8,9)>;
G:=Group( (1,5,3)(2,8,9)(4,7,6), (1,4,2)(3,6,9)(5,7,8), (2,3,4,5)(6,7,8,9), (3,5)(6,7)(8,9) );
G=PermutationGroup([[(1,5,3),(2,8,9),(4,7,6)], [(1,4,2),(3,6,9),(5,7,8)], [(2,3,4,5),(6,7,8,9)], [(3,5),(6,7),(8,9)]])
G:=TransitiveGroup(9,16);
►On 12 points - transitive group 12T34
Generators in S12
(1 7 5)(2 8 6)(3 9 11)(4 10 12)
(1 7 5)(2 6 8)(3 9 11)(4 12 10)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 10)(6 9)(7 12)(8 11)
G:=sub<Sym(12)| (1,7,5)(2,8,6)(3,9,11)(4,10,12), (1,7,5)(2,6,8)(3,9,11)(4,12,10), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,10)(6,9)(7,12)(8,11)>;
G:=Group( (1,7,5)(2,8,6)(3,9,11)(4,10,12), (1,7,5)(2,6,8)(3,9,11)(4,12,10), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,10)(6,9)(7,12)(8,11) );
G=PermutationGroup([[(1,7,5),(2,8,6),(3,9,11),(4,10,12)], [(1,7,5),(2,6,8),(3,9,11),(4,12,10)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,4),(2,3),(5,10),(6,9),(7,12),(8,11)]])
G:=TransitiveGroup(12,34);
►On 12 points - transitive group 12T35
Generators in S12
(1 9 8)(2 10 5)(3 6 11)(4 7 12)
(1 9 8)(2 5 10)(3 6 11)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)
G:=sub<Sym(12)| (1,9,8)(2,10,5)(3,6,11)(4,7,12), (1,9,8)(2,5,10)(3,6,11)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)>;
G:=Group( (1,9,8)(2,10,5)(3,6,11)(4,7,12), (1,9,8)(2,5,10)(3,6,11)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11) );
G=PermutationGroup([[(1,9,8),(2,10,5),(3,6,11),(4,7,12)], [(1,9,8),(2,5,10),(3,6,11),(4,12,7)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]])
G:=TransitiveGroup(12,35);
►On 12 points - transitive group 12T36
Generators in S12
(1 9 8)(3 6 11)
(2 5 10)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 12)(6 11)(7 10)(8 9)
G:=sub<Sym(12)| (1,9,8)(3,6,11), (2,5,10)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,12)(6,11)(7,10)(8,9)>;
G:=Group( (1,9,8)(3,6,11), (2,5,10)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,12)(6,11)(7,10)(8,9) );
G=PermutationGroup([[(1,9,8),(3,6,11)], [(2,5,10),(4,12,7)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(2,4),(5,12),(6,11),(7,10),(8,9)]])
G:=TransitiveGroup(12,36);
►On 18 points - transitive group 18T34
Generators in S18
(1 3 5)(2 13 11)(4 17 8)(6 10 15)(7 18 14)(9 12 16)
(1 12 14)(2 6 4)(3 16 7)(5 9 18)(8 11 15)(10 17 13)
(1 2)(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 11)(4 14)(5 13)(6 12)(7 8)(9 10)(15 16)(17 18)
G:=sub<Sym(18)| (1,3,5)(2,13,11)(4,17,8)(6,10,15)(7,18,14)(9,12,16), (1,12,14)(2,6,4)(3,16,7)(5,9,18)(8,11,15)(10,17,13), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,11)(4,14)(5,13)(6,12)(7,8)(9,10)(15,16)(17,18)>;
G:=Group( (1,3,5)(2,13,11)(4,17,8)(6,10,15)(7,18,14)(9,12,16), (1,12,14)(2,6,4)(3,16,7)(5,9,18)(8,11,15)(10,17,13), (1,2)(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,11)(4,14)(5,13)(6,12)(7,8)(9,10)(15,16)(17,18) );
G=PermutationGroup([[(1,3,5),(2,13,11),(4,17,8),(6,10,15),(7,18,14),(9,12,16)], [(1,12,14),(2,6,4),(3,16,7),(5,9,18),(8,11,15),(10,17,13)], [(1,2),(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(1,2),(3,11),(4,14),(5,13),(6,12),(7,8),(9,10),(15,16),(17,18)]])
G:=TransitiveGroup(18,34);
►On 18 points - transitive group 18T36
Generators in S18
(1 11 13)(2 4 6)(3 7 10)(5 8 9)(12 15 16)(14 18 17)
(1 14 12)(2 3 5)(4 7 8)(6 10 9)(11 18 15)(13 17 16)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 14)(4 13)(5 12)(6 11)(7 17)(8 16)(9 15)(10 18)
G:=sub<Sym(18)| (1,11,13)(2,4,6)(3,7,10)(5,8,9)(12,15,16)(14,18,17), (1,14,12)(2,3,5)(4,7,8)(6,10,9)(11,18,15)(13,17,16), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,14)(4,13)(5,12)(6,11)(7,17)(8,16)(9,15)(10,18)>;
G:=Group( (1,11,13)(2,4,6)(3,7,10)(5,8,9)(12,15,16)(14,18,17), (1,14,12)(2,3,5)(4,7,8)(6,10,9)(11,18,15)(13,17,16), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,14)(4,13)(5,12)(6,11)(7,17)(8,16)(9,15)(10,18) );
G=PermutationGroup([[(1,11,13),(2,4,6),(3,7,10),(5,8,9),(12,15,16),(14,18,17)], [(1,14,12),(2,3,5),(4,7,8),(6,10,9),(11,18,15),(13,17,16)], [(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(1,2),(3,14),(4,13),(5,12),(6,11),(7,17),(8,16),(9,15),(10,18)]])
G:=TransitiveGroup(18,36);
►On 24 points - transitive group 24T72
Generators in S24
(1 21 20)(2 22 17)(3 18 23)(4 19 24)(5 14 10)(6 15 11)(7 12 16)(8 9 13)
(1 21 20)(2 17 22)(3 18 23)(4 24 19)(5 14 10)(6 11 15)(7 12 16)(8 13 9)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
G:=sub<Sym(24)| (1,21,20)(2,22,17)(3,18,23)(4,19,24)(5,14,10)(6,15,11)(7,12,16)(8,9,13), (1,21,20)(2,17,22)(3,18,23)(4,24,19)(5,14,10)(6,11,15)(7,12,16)(8,13,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;
G:=Group( (1,21,20)(2,22,17)(3,18,23)(4,19,24)(5,14,10)(6,15,11)(7,12,16)(8,9,13), (1,21,20)(2,17,22)(3,18,23)(4,24,19)(5,14,10)(6,11,15)(7,12,16)(8,13,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );
G=PermutationGroup([[(1,21,20),(2,22,17),(3,18,23),(4,19,24),(5,14,10),(6,15,11),(7,12,16),(8,9,13)], [(1,21,20),(2,17,22),(3,18,23),(4,24,19),(5,14,10),(6,11,15),(7,12,16),(8,13,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)]])
G:=TransitiveGroup(24,72);
S3≀C2 is a maximal subgroup of
AΓL1(𝔽9) C33⋊D4 C32⋊2D12 S32⋊D5 C32⋊D20
S3≀C2 is a maximal quotient of
S32⋊C4 C3⋊S3.Q8 C32⋊D8 C32⋊2SD16 C32⋊Q16 He3⋊D4 C33⋊D4 C32⋊2D12 S32⋊D5 C32⋊D20
| action | f(x) | Disc(f) |
|---|---|---|
| 6T13 | x6-2x5+x4+1 | -26·5·149 |
| 9T16 | x9-10x7-3x6+25x5+5x4-21x3+5x-1 | 26·193·1373 |
| 12T34 | x12+12x10+54x8+104x6+57x4-36x2+16 | 256·320 |
| 12T35 | x12-4x10+12x8-51x6+164x4-8x2+433 | 212·36·194·314·4333·268214 |
| 12T36 | x12-x9-x6-x3+1 | -315·136 |
Matrix representation of S3≀C2 ►in GL4(ℤ) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | -1 |
| 0 | 1 | 0 | 0 |
| -1 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 1 | 1 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,Integers())| [1,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,-1],[0,-1,0,0,1,-1,0,0,0,0,1,0,0,0,0,1],[0,0,-1,0,0,0,0,-1,-1,1,0,0,0,1,0,0],[-1,0,0,0,0,-1,0,0,0,0,-1,1,0,0,0,1] >;
S3≀C2 in GAP, Magma, Sage, TeX
S_3\wr C_2
% in TeX
G:=Group("S3wrC2"); // GroupNames label
G:=SmallGroup(72,40);
// by ID
G=gap.SmallGroup(72,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,3,61,323,248,93,109,314]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of S3≀C2 in TeX
Character table of S3≀C2 in TeX