direct product, non-abelian, supersoluble, monomial
Aliases: S3×He3⋊C2, He3⋊8D6, C33⋊5D6, C32⋊4S32, (S3×He3)⋊2C2, (S3×C32)⋊2S3, (C3×He3)⋊4C22, He3⋊5S3⋊1C2, C3.4(S3×C3⋊S3), C3⋊1(C2×He3⋊C2), C32.8(C2×C3⋊S3), (C3×S3).2(C3⋊S3), (C3×He3⋊C2)⋊2C2, SmallGroup(324,122)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×He3 — S3×He3⋊C2 |
Generators and relations for S3×He3⋊C2
G = < a,b,c,d,e,f | a3=b2=c3=d3=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, df=fd, fef=e-1 >
Subgroups: 820 in 144 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C3, C3, C22, S3, S3, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3×S3, C3⋊S3, C3×C6, He3, He3, C33, S32, S3×C6, He3⋊C2, He3⋊C2, C2×He3, S3×C32, S3×C32, C3×C3⋊S3, C3×He3, C2×He3⋊C2, C3×S32, S3×He3, C3×He3⋊C2, He3⋊5S3, S3×He3⋊C2
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S32, C2×C3⋊S3, He3⋊C2, C2×He3⋊C2, S3×C3⋊S3, S3×He3⋊C2
Character table of S3×He3⋊C2
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 3M | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | |
| size | 1 | 3 | 9 | 27 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 3 | 3 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 27 | 27 | |
| 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 | 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 | -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 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ7 | 2 | 0 | -2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ9 | 2 | 0 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ15 | 3 | 3 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | ζ32 | ζ3 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 1 | ζ32 | ζ3 | complex lifted from He3⋊C2 |
| ρ16 | 3 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1 | ζ6 | ζ65 | complex lifted from He3⋊C2 |
| ρ17 | 3 | -3 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | ζ3 | ζ32 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 1 | ζ65 | ζ6 | complex lifted from C2×He3⋊C2 |
| ρ18 | 3 | 3 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | ζ3 | ζ32 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 1 | ζ3 | ζ32 | complex lifted from He3⋊C2 |
| ρ19 | 3 | -3 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1 | ζ3 | ζ32 | complex lifted from C2×He3⋊C2 |
| ρ20 | 3 | -3 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | ζ32 | ζ3 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 1 | ζ6 | ζ65 | complex lifted from C2×He3⋊C2 |
| ρ21 | 3 | -3 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1 | ζ32 | ζ3 | complex lifted from C2×He3⋊C2 |
| ρ22 | 3 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1 | ζ65 | ζ6 | complex lifted from He3⋊C2 |
| ρ23 | 4 | 0 | 0 | 0 | 4 | 4 | -2 | -2 | -2 | -2 | 4 | -2 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ24 | 4 | 0 | 0 | 0 | 4 | 4 | -2 | -2 | -2 | 4 | -2 | -2 | -2 | 1 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ25 | 4 | 0 | 0 | 0 | 4 | 4 | -2 | -2 | -2 | -2 | -2 | 4 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ26 | 4 | 0 | 0 | 0 | 4 | 4 | -2 | -2 | -2 | -2 | -2 | -2 | 4 | 1 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ27 | 6 | 0 | -2 | 0 | -3+3√-3 | -3-3√-3 | 3-3√-3/2 | -3 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | complex faithful |
| ρ28 | 6 | 0 | 2 | 0 | -3+3√-3 | -3-3√-3 | 3-3√-3/2 | -3 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | complex faithful |
| ρ29 | 6 | 0 | 2 | 0 | -3-3√-3 | -3+3√-3 | 3+3√-3/2 | -3 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | complex faithful |
| ρ30 | 6 | 0 | -2 | 0 | -3-3√-3 | -3+3√-3 | 3+3√-3/2 | -3 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | complex faithful |
►On 18 points - transitive group 18T135
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 8 5)(2 9 6)(3 7 4)(10 16 13)(11 17 14)(12 18 15)
(1 2 3)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 18 17)
(4 6 5)(7 8 9)(13 14 15)(16 18 17)
(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), (1,8,5)(2,9,6)(3,7,4)(10,16,13)(11,17,14)(12,18,15), (1,2,3)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,18,17), (4,6,5)(7,8,9)(13,14,15)(16,18,17), (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), (1,8,5)(2,9,6)(3,7,4)(10,16,13)(11,17,14)(12,18,15), (1,2,3)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,18,17), (4,6,5)(7,8,9)(13,14,15)(16,18,17), (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)], [(1,8,5),(2,9,6),(3,7,4),(10,16,13),(11,17,14),(12,18,15)], [(1,2,3),(4,5,6),(7,8,9),(10,12,11),(13,15,14),(16,18,17)], [(4,6,5),(7,8,9),(13,14,15),(16,18,17)], [(4,7),(5,8),(6,9),(13,16),(14,17),(15,18)]])
G:=TransitiveGroup(18,135);
►On 27 points - transitive group 27T118
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)(4 6)(7 8)(11 12)(14 15)(16 18)(20 21)(22 24)(25 27)
(1 9 23)(2 7 24)(3 8 22)(4 21 12)(5 19 10)(6 20 11)(13 26 17)(14 27 18)(15 25 16)
(1 10 13)(2 11 14)(3 12 15)(4 25 8)(5 26 9)(6 27 7)(16 22 21)(17 23 19)(18 24 20)
(1 26 17)(2 27 18)(3 25 16)(4 21 15)(5 19 13)(6 20 14)(7 24 11)(8 22 12)(9 23 10)
(4 21)(5 19)(6 20)(7 24)(8 22)(9 23)(16 25)(17 26)(18 27)
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)(4,6)(7,8)(11,12)(14,15)(16,18)(20,21)(22,24)(25,27), (1,9,23)(2,7,24)(3,8,22)(4,21,12)(5,19,10)(6,20,11)(13,26,17)(14,27,18)(15,25,16), (1,10,13)(2,11,14)(3,12,15)(4,25,8)(5,26,9)(6,27,7)(16,22,21)(17,23,19)(18,24,20), (1,26,17)(2,27,18)(3,25,16)(4,21,15)(5,19,13)(6,20,14)(7,24,11)(8,22,12)(9,23,10), (4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(16,25)(17,26)(18,27)>;
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)(4,6)(7,8)(11,12)(14,15)(16,18)(20,21)(22,24)(25,27), (1,9,23)(2,7,24)(3,8,22)(4,21,12)(5,19,10)(6,20,11)(13,26,17)(14,27,18)(15,25,16), (1,10,13)(2,11,14)(3,12,15)(4,25,8)(5,26,9)(6,27,7)(16,22,21)(17,23,19)(18,24,20), (1,26,17)(2,27,18)(3,25,16)(4,21,15)(5,19,13)(6,20,14)(7,24,11)(8,22,12)(9,23,10), (4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(16,25)(17,26)(18,27) );
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),(4,6),(7,8),(11,12),(14,15),(16,18),(20,21),(22,24),(25,27)], [(1,9,23),(2,7,24),(3,8,22),(4,21,12),(5,19,10),(6,20,11),(13,26,17),(14,27,18),(15,25,16)], [(1,10,13),(2,11,14),(3,12,15),(4,25,8),(5,26,9),(6,27,7),(16,22,21),(17,23,19),(18,24,20)], [(1,26,17),(2,27,18),(3,25,16),(4,21,15),(5,19,13),(6,20,14),(7,24,11),(8,22,12),(9,23,10)], [(4,21),(5,19),(6,20),(7,24),(8,22),(9,23),(16,25),(17,26),(18,27)]])
G:=TransitiveGroup(27,118);
Matrix representation of S3×He3⋊C2 ►in GL5(𝔽7)
| 6 | 1 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 6 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(5,GF(7))| [6,6,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,5,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,2],[6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,5,0] >;
S3×He3⋊C2 in GAP, Magma, Sage, TeX
S_3\times {\rm He}_3\rtimes C_2 % in TeX
G:=Group("S3xHe3:C2"); // GroupNames label
G:=SmallGroup(324,122);
// by ID
G=gap.SmallGroup(324,122);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,297,735,2164]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^3=d^3=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,c*d=d*c,e*c*e^-1=c*d^-1,f*c*f=c^-1,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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