non-abelian, supersoluble, monomial
Aliases: He3.1C6, (C3×C9)⋊2S3, He3⋊C2.C3, He3.C3⋊3C2, C32.2(C3×S3), C3.7(C32⋊C6), SmallGroup(162,12)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3.C6 |
Generators and relations for He3.C6
G = < a,b,c,d | a3=b3=c3=1, d6=b-1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >
Character table of He3.C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 9 | 1 | 1 | 6 | 18 | 9 | 9 | 3 | 3 | 3 | 3 | 3 | 3 | 18 | 18 | 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 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ10 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | ζ6 | ζ94+2ζ9 | ζ97+2ζ94 | 2ζ98+ζ95 | 2ζ95+ζ92 | ζ98+2ζ92 | 2ζ97+ζ9 | 0 | 0 | -ζ94 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ95 | complex faithful |
| ρ11 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ32 | ζ3 | 2ζ98+ζ95 | 2ζ95+ζ92 | ζ94+2ζ9 | ζ97+2ζ94 | 2ζ97+ζ9 | ζ98+2ζ92 | 0 | 0 | ζ95 | ζ97 | ζ9 | ζ98 | ζ92 | ζ94 | complex faithful |
| ρ12 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | ζ6 | ζ97+2ζ94 | 2ζ97+ζ9 | 2ζ95+ζ92 | ζ98+2ζ92 | 2ζ98+ζ95 | ζ94+2ζ9 | 0 | 0 | -ζ97 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ92 | complex faithful |
| ρ13 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | ζ65 | ζ98+2ζ92 | 2ζ98+ζ95 | 2ζ97+ζ9 | ζ94+2ζ9 | ζ97+2ζ94 | 2ζ95+ζ92 | 0 | 0 | -ζ98 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ9 | complex faithful |
| ρ14 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | ζ6 | 2ζ97+ζ9 | ζ94+2ζ9 | ζ98+2ζ92 | 2ζ98+ζ95 | 2ζ95+ζ92 | ζ97+2ζ94 | 0 | 0 | -ζ9 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ98 | complex faithful |
| ρ15 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | ζ65 | 2ζ98+ζ95 | 2ζ95+ζ92 | ζ94+2ζ9 | ζ97+2ζ94 | 2ζ97+ζ9 | ζ98+2ζ92 | 0 | 0 | -ζ95 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ94 | complex faithful |
| ρ16 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ3 | ζ32 | 2ζ97+ζ9 | ζ94+2ζ9 | ζ98+2ζ92 | 2ζ98+ζ95 | 2ζ95+ζ92 | ζ97+2ζ94 | 0 | 0 | ζ9 | ζ95 | ζ92 | ζ97 | ζ94 | ζ98 | complex faithful |
| ρ17 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ3 | ζ32 | ζ94+2ζ9 | ζ97+2ζ94 | 2ζ98+ζ95 | 2ζ95+ζ92 | ζ98+2ζ92 | 2ζ97+ζ9 | 0 | 0 | ζ94 | ζ92 | ζ98 | ζ9 | ζ97 | ζ95 | complex faithful |
| ρ18 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ32 | ζ3 | ζ98+2ζ92 | 2ζ98+ζ95 | 2ζ97+ζ9 | ζ94+2ζ9 | ζ97+2ζ94 | 2ζ95+ζ92 | 0 | 0 | ζ98 | ζ94 | ζ97 | ζ92 | ζ95 | ζ9 | complex faithful |
| ρ19 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ32 | ζ3 | 2ζ95+ζ92 | ζ98+2ζ92 | ζ97+2ζ94 | 2ζ97+ζ9 | ζ94+2ζ9 | 2ζ98+ζ95 | 0 | 0 | ζ92 | ζ9 | ζ94 | ζ95 | ζ98 | ζ97 | complex faithful |
| ρ20 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ3 | ζ32 | ζ97+2ζ94 | 2ζ97+ζ9 | 2ζ95+ζ92 | ζ98+2ζ92 | 2ζ98+ζ95 | ζ94+2ζ9 | 0 | 0 | ζ97 | ζ98 | ζ95 | ζ94 | ζ9 | ζ92 | complex faithful |
| ρ21 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | ζ65 | 2ζ95+ζ92 | ζ98+2ζ92 | ζ97+2ζ94 | 2ζ97+ζ9 | ζ94+2ζ9 | 2ζ98+ζ95 | 0 | 0 | -ζ92 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ97 | 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 27T45
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(11 23 17)(13 25 19)(15 27 21)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)
(1 25 16)(2 11 20)(3 15 24)(4 19 10)(5 23 14)(6 27 18)(7 13 22)(8 17 26)(9 21 12)
(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)| (1,7,4)(2,8,5)(3,9,6)(11,23,17)(13,25,19)(15,27,21), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,25,16)(2,11,20)(3,15,24)(4,19,10)(5,23,14)(6,27,18)(7,13,22)(8,17,26)(9,21,12), (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( (1,7,4)(2,8,5)(3,9,6)(11,23,17)(13,25,19)(15,27,21), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,25,16)(2,11,20)(3,15,24)(4,19,10)(5,23,14)(6,27,18)(7,13,22)(8,17,26)(9,21,12), (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([[(1,7,4),(2,8,5),(3,9,6),(11,23,17),(13,25,19),(15,27,21)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21)], [(1,25,16),(2,11,20),(3,15,24),(4,19,10),(5,23,14),(6,27,18),(7,13,22),(8,17,26),(9,21,12)], [(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(27,45);
►On 27 points - transitive group 27T69
Generators in S27
(1 19 16)(2 11 26)(3 21 18)(4 13 10)(5 23 20)(6 15 12)(7 25 22)(8 17 14)(9 27 24)
(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)
(1 19 10)(2 26 17)(4 13 22)(5 20 11)(7 25 16)(8 14 23)(12 18 24)(15 27 21)
(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)| (1,19,16)(2,11,26)(3,21,18)(4,13,10)(5,23,20)(6,15,12)(7,25,22)(8,17,14)(9,27,24), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,19,10)(2,26,17)(4,13,22)(5,20,11)(7,25,16)(8,14,23)(12,18,24)(15,27,21), (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( (1,19,16)(2,11,26)(3,21,18)(4,13,10)(5,23,20)(6,15,12)(7,25,22)(8,17,14)(9,27,24), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,19,10)(2,26,17)(4,13,22)(5,20,11)(7,25,16)(8,14,23)(12,18,24)(15,27,21), (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([[(1,19,16),(2,11,26),(3,21,18),(4,13,10),(5,23,20),(6,15,12),(7,25,22),(8,17,14),(9,27,24)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21)], [(1,19,10),(2,26,17),(4,13,22),(5,20,11),(7,25,16),(8,14,23),(12,18,24),(15,27,21)], [(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(27,69);
He3.C6 is a maximal subgroup of
He3.D6 C3≀S3⋊3C3 He3.C3⋊C6 C3≀C3.C6 He3.C3⋊S3
He3.C6 is a maximal quotient of He3.C12 C32⋊C9⋊S3 C33.(C3×S3) C32⋊2D9.C3 (C3×C9)⋊D9 He3⋊C18 He3.C3⋊S3
Matrix representation of He3.C6 ►in GL3(𝔽19) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 7 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 1 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 7 |
| 18 | 8 | 18 |
| 18 | 12 | 12 |
| 8 | 8 | 12 |
G:=sub<GL(3,GF(19))| [0,0,1,1,0,0,0,1,0],[7,0,0,0,7,0,0,0,7],[1,0,0,0,11,0,0,0,7],[18,18,8,8,12,8,18,12,12] >;
He3.C6 in GAP, Magma, Sage, TeX
{\rm He}_3.C_6 % in TeX
G:=Group("He3.C6"); // GroupNames label
G:=SmallGroup(162,12);
// by ID
G=gap.SmallGroup(162,12);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,276,182,187,2883,433]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^6=b^-1,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.C6 in TeX
Character table of He3.C6 in TeX