direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C9, C3⋊C18, C32.2C6, (C3×C9)⋊1C2, (C3×S3).C3, C3.4(C3×S3), SmallGroup(54,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C9 |
Generators and relations for S3×C9
G = < a,b,c | a9=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Character table of S3×C9
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | |
| 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 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ95 | ζ97 | ζ98 | ζ92 | ζ9 | ζ94 | ζ95 | ζ9 | ζ92 | ζ97 | ζ94 | ζ98 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | ζ92 | linear of order 9 |
| ρ8 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ98 | ζ94 | ζ92 | ζ95 | ζ97 | ζ9 | ζ98 | ζ97 | ζ95 | ζ94 | ζ9 | ζ92 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | ζ95 | linear of order 9 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ9 | ζ95 | ζ97 | ζ94 | ζ92 | ζ98 | ζ9 | ζ92 | ζ94 | ζ95 | ζ98 | ζ97 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | ζ94 | linear of order 9 |
| ρ10 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | ζ92 | ζ9 | ζ95 | ζ98 | ζ94 | ζ97 | ζ92 | ζ94 | ζ98 | ζ9 | ζ97 | ζ95 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | ζ98 | linear of order 9 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ94 | ζ92 | ζ9 | ζ97 | ζ98 | ζ95 | ζ94 | ζ98 | ζ97 | ζ92 | ζ95 | ζ9 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ97 | linear of order 18 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ97 | ζ98 | ζ94 | ζ9 | ζ95 | ζ92 | ζ97 | ζ95 | ζ9 | ζ98 | ζ92 | ζ94 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ9 | linear of order 18 |
| ρ13 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ92 | ζ9 | ζ95 | ζ98 | ζ94 | ζ97 | ζ92 | ζ94 | ζ98 | ζ9 | ζ97 | ζ95 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ98 | linear of order 18 |
| ρ14 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ97 | ζ98 | ζ94 | ζ9 | ζ95 | ζ92 | ζ97 | ζ95 | ζ9 | ζ98 | ζ92 | ζ94 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | ζ9 | linear of order 9 |
| ρ15 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ65 | ζ6 | ζ9 | ζ95 | ζ97 | ζ94 | ζ92 | ζ98 | ζ9 | ζ92 | ζ94 | ζ95 | ζ98 | ζ97 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ94 | linear of order 18 |
| ρ16 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ98 | ζ94 | ζ92 | ζ95 | ζ97 | ζ9 | ζ98 | ζ97 | ζ95 | ζ94 | ζ9 | ζ92 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ95 | linear of order 18 |
| ρ17 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | ζ94 | ζ92 | ζ9 | ζ97 | ζ98 | ζ95 | ζ94 | ζ98 | ζ97 | ζ92 | ζ95 | ζ9 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | ζ97 | linear of order 9 |
| ρ18 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ6 | ζ65 | ζ95 | ζ97 | ζ98 | ζ92 | ζ9 | ζ94 | ζ95 | ζ9 | ζ92 | ζ97 | ζ94 | ζ98 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ92 | linear of order 18 |
| ρ19 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ20 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ21 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ22 | 2 | 0 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 2ζ95 | 2ζ97 | 2ζ98 | 2ζ92 | 2ζ9 | 2ζ94 | -ζ95 | -ζ9 | -ζ92 | -ζ97 | -ζ94 | -ζ98 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 2 | 0 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 2ζ92 | 2ζ9 | 2ζ95 | 2ζ98 | 2ζ94 | 2ζ97 | -ζ92 | -ζ94 | -ζ98 | -ζ9 | -ζ97 | -ζ95 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 2 | 0 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 2ζ94 | 2ζ92 | 2ζ9 | 2ζ97 | 2ζ98 | 2ζ95 | -ζ94 | -ζ98 | -ζ97 | -ζ92 | -ζ95 | -ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 2 | 0 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | 0 | 2ζ98 | 2ζ94 | 2ζ92 | 2ζ95 | 2ζ97 | 2ζ9 | -ζ98 | -ζ97 | -ζ95 | -ζ94 | -ζ9 | -ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 2 | 0 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 2ζ9 | 2ζ95 | 2ζ97 | 2ζ94 | 2ζ92 | 2ζ98 | -ζ9 | -ζ92 | -ζ94 | -ζ95 | -ζ98 | -ζ97 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 2 | 0 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | 0 | 2ζ97 | 2ζ98 | 2ζ94 | 2ζ9 | 2ζ95 | 2ζ92 | -ζ97 | -ζ95 | -ζ9 | -ζ98 | -ζ92 | -ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 18 points - transitive group 18T16
Generators in S18
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)
G:=sub<Sym(18)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)>;
G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13)]])
G:=TransitiveGroup(18,16);
►On 27 points - transitive group 27T12
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)
(1 14 24)(2 15 25)(3 16 26)(4 17 27)(5 18 19)(6 10 20)(7 11 21)(8 12 22)(9 13 23)
(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)
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), (1,14,24)(2,15,25)(3,16,26)(4,17,27)(5,18,19)(6,10,20)(7,11,21)(8,12,22)(9,13,23), (10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19)>;
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), (1,14,24)(2,15,25)(3,16,26)(4,17,27)(5,18,19)(6,10,20)(7,11,21)(8,12,22)(9,13,23), (10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19) );
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)], [(1,14,24),(2,15,25),(3,16,26),(4,17,27),(5,18,19),(6,10,20),(7,11,21),(8,12,22),(9,13,23)], [(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,19)]])
G:=TransitiveGroup(27,12);
S3×C9 is a maximal subgroup of
C32⋊C18 C9⋊C18 He3.C6 He3.2C6 He3.4C6 D21⋊C9
S3×C9 is a maximal quotient of C32⋊C18 C9⋊C18 D21⋊C9
Matrix representation of S3×C9 ►in GL2(𝔽19) generated by
| 6 | 0 |
| 0 | 6 |
| 7 | 0 |
| 0 | 11 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(19))| [6,0,0,6],[7,0,0,11],[0,1,1,0] >;
S3×C9 in GAP, Magma, Sage, TeX
S_3\times C_9
% in TeX
G:=Group("S3xC9"); // GroupNames label
G:=SmallGroup(54,4);
// by ID
G=gap.SmallGroup(54,4);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,29,579]);
// Polycyclic
G:=Group<a,b,c|a^9=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of S3×C9 in TeX
Character table of S3×C9 in TeX