direct product, non-abelian, soluble, monomial
Aliases: C3×S3≀C2, C33⋊1D4, S32⋊C6, C32⋊C4⋊C6, C32⋊(C3×D4), (C3×S32)⋊1C2, C3⋊S3.1(C2×C6), (C3×C32⋊C4)⋊3C2, (C3×C3⋊S3).1C22, SmallGroup(216,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×S3≀C2
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >
Subgroups: 272 in 60 conjugacy classes, 14 normal (10 characteristic)
C1, C2, C3, C3, C4, C22, S3, C6, D4, C32, C32, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×D4, C33, C32⋊C4, S32, S3×C6, S3×C32, C3×C3⋊S3, S3≀C2, C3×C32⋊C4, C3×S32, C3×S3≀C2
Quotients: C1, C2, C3, C22, C6, D4, C2×C6, C3×D4, S3≀C2, C3×S3≀C2
Character table of C3×S3≀C2
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 12A | 12B | |
| size | 1 | 6 | 6 | 9 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 18 | 6 | 6 | 6 | 6 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 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 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | -1 | ζ3 | ζ6 | ζ65 | ζ32 | ζ32 | ζ3 | -1 | ζ65 | ζ32 | ζ6 | ζ3 | 1 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | -1 | ζ6 | ζ3 | ζ32 | ζ65 | ζ3 | ζ32 | 1 | ζ32 | ζ65 | ζ3 | ζ6 | -1 | ζ65 | ζ6 | linear of order 6 |
| ρ8 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | -1 | ζ32 | ζ65 | ζ6 | ζ3 | ζ3 | ζ32 | -1 | ζ6 | ζ3 | ζ65 | ζ32 | 1 | ζ65 | ζ6 | linear of order 6 |
| ρ9 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | -1 | ζ65 | ζ32 | ζ3 | ζ6 | ζ32 | ζ3 | 1 | ζ3 | ζ6 | ζ32 | ζ65 | -1 | ζ6 | ζ65 | linear of order 6 |
| ρ10 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | -1 | ζ32 | ζ3 | linear of order 6 |
| ρ11 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | -1 | ζ3 | ζ32 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ13 | 2 | 0 | 0 | -2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 0 | 0 | -2 | -1-√-3 | -1+√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ15 | 2 | 0 | 0 | -2 | -1+√-3 | -1-√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ16 | 4 | 0 | -2 | 0 | 4 | 4 | 1 | -2 | -2 | 1 | -2 | 1 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | 0 | 2 | 0 | 4 | 4 | 1 | -2 | -2 | 1 | -2 | 1 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ18 | 4 | -2 | 0 | 0 | 4 | 4 | -2 | 1 | 1 | -2 | 1 | -2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ19 | 4 | 2 | 0 | 0 | 4 | 4 | -2 | 1 | 1 | -2 | 1 | -2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ20 | 4 | 2 | 0 | 0 | -2+2√-3 | -2-2√-3 | 1+√-3 | ζ32 | ζ3 | 1-√-3 | 1 | -2 | 0 | -1+√-3 | 0 | 0 | -1-√-3 | 0 | 0 | 0 | 0 | ζ6 | 0 | ζ65 | -1 | 0 | 0 | complex faithful |
| ρ21 | 4 | 0 | 2 | 0 | -2+2√-3 | -2-2√-3 | ζ32 | 1+√-3 | 1-√-3 | ζ3 | -2 | 1 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | -1 | ζ65 | 0 | ζ6 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -2 | 0 | 0 | -2-2√-3 | -2+2√-3 | 1-√-3 | ζ3 | ζ32 | 1+√-3 | 1 | -2 | 0 | 1+√-3 | 0 | 0 | 1-√-3 | 0 | 0 | 0 | 0 | ζ3 | 0 | ζ32 | 1 | 0 | 0 | complex faithful |
| ρ23 | 4 | 0 | -2 | 0 | -2-2√-3 | -2+2√-3 | ζ3 | 1-√-3 | 1+√-3 | ζ32 | -2 | 1 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 1 | ζ32 | 0 | ζ3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -2 | 0 | 0 | -2+2√-3 | -2-2√-3 | 1+√-3 | ζ32 | ζ3 | 1-√-3 | 1 | -2 | 0 | 1-√-3 | 0 | 0 | 1+√-3 | 0 | 0 | 0 | 0 | ζ32 | 0 | ζ3 | 1 | 0 | 0 | complex faithful |
| ρ25 | 4 | 0 | 2 | 0 | -2-2√-3 | -2+2√-3 | ζ3 | 1-√-3 | 1+√-3 | ζ32 | -2 | 1 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | -1 | ζ6 | 0 | ζ65 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 4 | 0 | -2 | 0 | -2+2√-3 | -2-2√-3 | ζ32 | 1+√-3 | 1-√-3 | ζ3 | -2 | 1 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 1 | ζ3 | 0 | ζ32 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | 2 | 0 | 0 | -2-2√-3 | -2+2√-3 | 1-√-3 | ζ3 | ζ32 | 1+√-3 | 1 | -2 | 0 | -1-√-3 | 0 | 0 | -1+√-3 | 0 | 0 | 0 | 0 | ζ65 | 0 | ζ6 | -1 | 0 | 0 | complex faithful |
►On 12 points - transitive group 12T121
Generators in S12
(1 9 8)(2 10 5)(3 11 6)(4 12 7)
(2 5 10)(4 12 7)
(1 8 9)(3 11 6)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 7)(10 12)
G:=sub<Sym(12)| (1,9,8)(2,10,5)(3,11,6)(4,12,7), (2,5,10)(4,12,7), (1,8,9)(3,11,6), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,7)(10,12)>;
G:=Group( (1,9,8)(2,10,5)(3,11,6)(4,12,7), (2,5,10)(4,12,7), (1,8,9)(3,11,6), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,7)(10,12) );
G=PermutationGroup([[(1,9,8),(2,10,5),(3,11,6),(4,12,7)], [(2,5,10),(4,12,7)], [(1,8,9),(3,11,6)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(2,4),(5,7),(10,12)]])
G:=TransitiveGroup(12,121);
►On 18 points - transitive group 18T93
Generators in S18
(1 3 6)(2 4 5)(7 18 11)(8 15 12)(9 16 13)(10 17 14)
(1 13 11)(2 14 12)(3 9 7)(4 10 8)(5 17 15)(6 16 18)
(1 13 11)(2 12 14)(3 9 7)(4 8 10)(5 15 17)(6 16 18)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 4)(5 6)(7 10)(8 9)(11 14)(12 13)(15 16)(17 18)
G:=sub<Sym(18)| (1,3,6)(2,4,5)(7,18,11)(8,15,12)(9,16,13)(10,17,14), (1,13,11)(2,14,12)(3,9,7)(4,10,8)(5,17,15)(6,16,18), (1,13,11)(2,12,14)(3,9,7)(4,8,10)(5,15,17)(6,16,18), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,4)(5,6)(7,10)(8,9)(11,14)(12,13)(15,16)(17,18)>;
G:=Group( (1,3,6)(2,4,5)(7,18,11)(8,15,12)(9,16,13)(10,17,14), (1,13,11)(2,14,12)(3,9,7)(4,10,8)(5,17,15)(6,16,18), (1,13,11)(2,12,14)(3,9,7)(4,8,10)(5,15,17)(6,16,18), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,4)(5,6)(7,10)(8,9)(11,14)(12,13)(15,16)(17,18) );
G=PermutationGroup([[(1,3,6),(2,4,5),(7,18,11),(8,15,12),(9,16,13),(10,17,14)], [(1,13,11),(2,14,12),(3,9,7),(4,10,8),(5,17,15),(6,16,18)], [(1,13,11),(2,12,14),(3,9,7),(4,8,10),(5,15,17),(6,16,18)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13),(15,16),(17,18)]])
G:=TransitiveGroup(18,93);
►On 24 points - transitive group 24T561
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(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)| (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (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( (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (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([[(1,20,21),(2,17,22),(3,18,23),(4,19,24),(5,10,14),(6,11,15),(7,12,16),(8,9,13)], [(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(24,561);
►On 27 points - transitive group 27T84
Generators in S27
(1 3 2)(4 10 13)(5 11 14)(6 8 15)(7 9 12)(16 20 27)(17 21 24)(18 22 25)(19 23 26)
(1 21 23)(2 17 19)(3 24 26)(4 5 22)(6 20 7)(8 27 9)(10 11 25)(12 15 16)(13 14 18)
(1 20 22)(2 16 18)(3 27 25)(4 21 7)(5 23 6)(8 11 26)(9 10 24)(12 13 17)(14 19 15)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
(4 5)(6 7)(8 9)(10 11)(12 15)(13 14)(17 19)(21 23)(24 26)
G:=sub<Sym(27)| (1,3,2)(4,10,13)(5,11,14)(6,8,15)(7,9,12)(16,20,27)(17,21,24)(18,22,25)(19,23,26), (1,21,23)(2,17,19)(3,24,26)(4,5,22)(6,20,7)(8,27,9)(10,11,25)(12,15,16)(13,14,18), (1,20,22)(2,16,18)(3,27,25)(4,21,7)(5,23,6)(8,11,26)(9,10,24)(12,13,17)(14,19,15), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,5)(6,7)(8,9)(10,11)(12,15)(13,14)(17,19)(21,23)(24,26)>;
G:=Group( (1,3,2)(4,10,13)(5,11,14)(6,8,15)(7,9,12)(16,20,27)(17,21,24)(18,22,25)(19,23,26), (1,21,23)(2,17,19)(3,24,26)(4,5,22)(6,20,7)(8,27,9)(10,11,25)(12,15,16)(13,14,18), (1,20,22)(2,16,18)(3,27,25)(4,21,7)(5,23,6)(8,11,26)(9,10,24)(12,13,17)(14,19,15), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,5)(6,7)(8,9)(10,11)(12,15)(13,14)(17,19)(21,23)(24,26) );
G=PermutationGroup([[(1,3,2),(4,10,13),(5,11,14),(6,8,15),(7,9,12),(16,20,27),(17,21,24),(18,22,25),(19,23,26)], [(1,21,23),(2,17,19),(3,24,26),(4,5,22),(6,20,7),(8,27,9),(10,11,25),(12,15,16),(13,14,18)], [(1,20,22),(2,16,18),(3,27,25),(4,21,7),(5,23,6),(8,11,26),(9,10,24),(12,13,17),(14,19,15)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,5),(6,7),(8,9),(10,11),(12,15),(13,14),(17,19),(21,23),(24,26)]])
G:=TransitiveGroup(27,84);
C3×S3≀C2 is a maximal subgroup of
C33⋊SD16
| action | f(x) | Disc(f) |
|---|---|---|
| 12T121 | x12-x9+2x6-2x3+1 | 318·133 |
Matrix representation of C3×S3≀C2 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 2 | 5 | 0 |
| 2 | 3 | 5 | 2 |
| 2 | 2 | 6 | 2 |
| 2 | 4 | 3 | 2 |
| 3 | 5 | 5 | 1 |
| 3 | 1 | 1 | 5 |
| 6 | 4 | 6 | 3 |
| 1 | 6 | 6 | 5 |
| 2 | 4 | 6 | 3 |
| 4 | 2 | 6 | 5 |
| 4 | 3 | 5 | 1 |
| 1 | 6 | 6 | 5 |
| 4 | 2 | 5 | 3 |
| 3 | 3 | 3 | 6 |
| 3 | 4 | 2 | 6 |
| 2 | 5 | 2 | 3 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[3,3,6,1,5,1,4,6,5,1,6,6,1,5,3,5],[2,4,4,1,4,2,3,6,6,6,5,6,3,5,1,5],[4,3,3,2,2,3,4,5,5,3,2,2,3,6,6,3] >;
C3×S3≀C2 in GAP, Magma, Sage, TeX
C_3\times S_3\wr C_2
% in TeX
G:=Group("C3xS3wrC2"); // GroupNames label
G:=SmallGroup(216,157);
// by ID
G=gap.SmallGroup(216,157);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,3,169,1444,1090,142,443,455]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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