non-abelian, supersoluble, monomial
Aliases: He3.2C6, (C3×C9)⋊3S3, He3⋊C3⋊1C2, He3⋊C2⋊2C3, C32.3(C3×S3), C3.8(C32⋊C6), SmallGroup(162,14)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3.2C6 |
Generators and relations for He3.2C6
G = < a,b,c,d | a3=b3=c3=1, d6=b, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >
Character table of He3.2C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 9 | 1 | 1 | 6 | 18 | 18 | 18 | 9 | 9 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | 2 | ζ6 | -1 | ζ65 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | 2 | 2 | ζ65 | -1 | ζ6 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ10 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 2ζ98+ζ92 | 2ζ94+ζ9 | 2ζ97+ζ94 | ζ97+2ζ9 | ζ95+2ζ92 | ζ98+2ζ95 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ92 | complex faithful |
| ρ11 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ98+2ζ95 | 2ζ97+ζ94 | ζ97+2ζ9 | 2ζ94+ζ9 | 2ζ98+ζ92 | ζ95+2ζ92 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | ζ98 | complex faithful |
| ρ12 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ98+2ζ95 | 2ζ97+ζ94 | ζ97+2ζ9 | 2ζ94+ζ9 | 2ζ98+ζ92 | ζ95+2ζ92 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ98 | complex faithful |
| ρ13 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 2ζ94+ζ9 | ζ95+2ζ92 | 2ζ98+ζ92 | ζ98+2ζ95 | ζ97+2ζ9 | 2ζ97+ζ94 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ9 | complex faithful |
| ρ14 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 2ζ97+ζ94 | 2ζ98+ζ92 | ζ98+2ζ95 | ζ95+2ζ92 | 2ζ94+ζ9 | ζ97+2ζ9 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ94 | complex faithful |
| ρ15 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ95+2ζ92 | ζ97+2ζ9 | 2ζ94+ζ9 | 2ζ97+ζ94 | ζ98+2ζ95 | 2ζ98+ζ92 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ95 | complex faithful |
| ρ16 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ97+2ζ9 | ζ98+2ζ95 | ζ95+2ζ92 | 2ζ98+ζ92 | 2ζ97+ζ94 | 2ζ94+ζ9 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ97 | complex faithful |
| ρ17 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 2ζ98+ζ92 | 2ζ94+ζ9 | 2ζ97+ζ94 | ζ97+2ζ9 | ζ95+2ζ92 | ζ98+2ζ95 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | ζ92 | complex faithful |
| ρ18 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ97+2ζ9 | ζ98+2ζ95 | ζ95+2ζ92 | 2ζ98+ζ92 | 2ζ97+ζ94 | 2ζ94+ζ9 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | ζ97 | complex faithful |
| ρ19 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ95+2ζ92 | ζ97+2ζ9 | 2ζ94+ζ9 | 2ζ97+ζ94 | ζ98+2ζ95 | 2ζ98+ζ92 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | ζ95 | complex faithful |
| ρ20 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 2ζ97+ζ94 | 2ζ98+ζ92 | ζ98+2ζ95 | ζ95+2ζ92 | 2ζ94+ζ9 | ζ97+2ζ9 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | ζ94 | complex faithful |
| ρ21 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 2ζ94+ζ9 | ζ95+2ζ92 | 2ζ98+ζ92 | ζ98+2ζ95 | ζ97+2ζ9 | 2ζ97+ζ94 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | ζ9 | complex faithful |
| ρ22 | 6 | 0 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
►On 27 points - transitive group 27T40
Generators in S27
(1 4 7)(2 5 8)(3 6 9)(11 17 23)(13 19 25)(15 21 27)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(1 25 16)(2 23 14)(3 21 12)(4 19 10)(5 17 26)(6 15 24)(7 13 22)(8 11 20)(9 27 18)
(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)
G:=sub<Sym(27)| (1,4,7)(2,5,8)(3,6,9)(11,17,23)(13,19,25)(15,21,27), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (1,25,16)(2,23,14)(3,21,12)(4,19,10)(5,17,26)(6,15,24)(7,13,22)(8,11,20)(9,27,18), (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)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(11,17,23)(13,19,25)(15,21,27), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (1,25,16)(2,23,14)(3,21,12)(4,19,10)(5,17,26)(6,15,24)(7,13,22)(8,11,20)(9,27,18), (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) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(11,17,23),(13,19,25),(15,21,27)], [(1,7,4),(2,8,5),(3,9,6),(10,16,22),(11,17,23),(12,18,24),(13,19,25),(14,20,26),(15,21,27)], [(1,25,16),(2,23,14),(3,21,12),(4,19,10),(5,17,26),(6,15,24),(7,13,22),(8,11,20),(9,27,18)], [(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)]])
G:=TransitiveGroup(27,40);
►On 27 points - transitive group 27T49
Generators in S27
(1 19 22)(2 11 14)(3 21 24)(4 13 16)(5 23 26)(6 15 18)(7 25 10)(8 17 20)(9 27 12)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(1 19 10)(2 14 23)(4 13 22)(5 26 17)(7 25 16)(8 20 11)(12 24 18)(15 21 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)
G:=sub<Sym(27)| (1,19,22)(2,11,14)(3,21,24)(4,13,16)(5,23,26)(6,15,18)(7,25,10)(8,17,20)(9,27,12), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (1,19,10)(2,14,23)(4,13,22)(5,26,17)(7,25,16)(8,20,11)(12,24,18)(15,21,27), (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)>;
G:=Group( (1,19,22)(2,11,14)(3,21,24)(4,13,16)(5,23,26)(6,15,18)(7,25,10)(8,17,20)(9,27,12), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (1,19,10)(2,14,23)(4,13,22)(5,26,17)(7,25,16)(8,20,11)(12,24,18)(15,21,27), (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) );
G=PermutationGroup([[(1,19,22),(2,11,14),(3,21,24),(4,13,16),(5,23,26),(6,15,18),(7,25,10),(8,17,20),(9,27,12)], [(1,7,4),(2,8,5),(3,9,6),(10,16,22),(11,17,23),(12,18,24),(13,19,25),(14,20,26),(15,21,27)], [(1,19,10),(2,14,23),(4,13,22),(5,26,17),(7,25,16),(8,20,11),(12,24,18),(15,21,27)], [(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)]])
G:=TransitiveGroup(27,49);
He3.2C6 is a maximal subgroup of
He3.2D6 C92⋊2S3 C3≀S3⋊3C3 He3.(C3×C6) C3≀C3.C6 He3⋊C3⋊2S3
He3.2C6 is a maximal quotient of
He3.2C12 C32⋊C9.S3 (C3×He3).C6 C32⋊C9.C6 (C3×C9)⋊3D9 He3⋊C18 C92⋊S3 C92.S3 C9⋊C9.S3 C9⋊C9.3S3 C9⋊C9⋊S3 He3⋊C3⋊2S3
Matrix representation of He3.2C6 ►in GL3(𝔽19) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 7 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 0 | 11 | 0 |
| 0 | 0 | 7 |
| 1 | 0 | 0 |
| 13 | 10 | 13 |
| 13 | 15 | 15 |
| 10 | 10 | 15 |
G:=sub<GL(3,GF(19))| [0,0,1,1,0,0,0,1,0],[7,0,0,0,7,0,0,0,7],[0,0,1,11,0,0,0,7,0],[13,13,10,10,15,10,13,15,15] >;
He3.2C6 in GAP, Magma, Sage, TeX
{\rm He}_3._2C_6 % in TeX
G:=Group("He3.2C6"); // GroupNames label
G:=SmallGroup(162,14);
// by ID
G=gap.SmallGroup(162,14);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,276,182,187,1803,253]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^6=b,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^-1>;
// generators/relations
Export
Subgroup lattice of He3.2C6 in TeX
Character table of He3.2C6 in TeX