p-group, metabelian, nilpotent (class 2), monomial
Aliases: He3, 3+ 1+2, C32⋊C3, C3.1C32, 3-Sylow(SL(3,3)), SmallGroup(27,3)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for He3
G = < a,b,c | a3=b3=c3=1, ab=ba, cac-1=ab-1, bc=cb >
Character table of He3
| class | 1 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | |
| size | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ11 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 9 points - transitive group 9T7
Generators in S9
(1 2 3)(4 5 6)(7 8 9)
(1 9 5)(2 7 6)(3 8 4)
(1 3 6)(2 9 8)(4 7 5)
G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4), (1,3,6)(2,9,8)(4,7,5)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4), (1,3,6)(2,9,8)(4,7,5) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9)], [(1,9,5),(2,7,6),(3,8,4)], [(1,3,6),(2,9,8),(4,7,5)]])
G:=TransitiveGroup(9,7);
►Regular action on 27 points - transitive group 27T3
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 13 11)(2 14 12)(3 15 10)(4 26 8)(5 27 9)(6 25 7)(16 19 22)(17 20 23)(18 21 24)
(1 5 24)(2 25 19)(3 8 17)(4 20 15)(6 16 12)(7 22 14)(9 21 11)(10 26 23)(13 27 18)
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), (1,13,11)(2,14,12)(3,15,10)(4,26,8)(5,27,9)(6,25,7)(16,19,22)(17,20,23)(18,21,24), (1,5,24)(2,25,19)(3,8,17)(4,20,15)(6,16,12)(7,22,14)(9,21,11)(10,26,23)(13,27,18)>;
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), (1,13,11)(2,14,12)(3,15,10)(4,26,8)(5,27,9)(6,25,7)(16,19,22)(17,20,23)(18,21,24), (1,5,24)(2,25,19)(3,8,17)(4,20,15)(6,16,12)(7,22,14)(9,21,11)(10,26,23)(13,27,18) );
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)], [(1,13,11),(2,14,12),(3,15,10),(4,26,8),(5,27,9),(6,25,7),(16,19,22),(17,20,23),(18,21,24)], [(1,5,24),(2,25,19),(3,8,17),(4,20,15),(6,16,12),(7,22,14),(9,21,11),(10,26,23),(13,27,18)]])
G:=TransitiveGroup(27,3);
He3 is a maximal subgroup of
C32⋊C6 He3⋊C2 C3≀C3 He3.C3 He3⋊C3 C9○He3 C32⋊A4 C7⋊He3 C13⋊He3
He3 is a maximal quotient of
C32⋊C9 C3≀C3 He3.C3 He3⋊C3 C3.He3 C32⋊A4 C7⋊He3 C13⋊He3
| action | f(x) | Disc(f) |
|---|---|---|
| 9T7 | x9-3x8-15x7+51x6+39x5-219x4+81x3+204x2-132x-8 | 26·316·194·22872 |
Matrix representation of He3 ►in GL3(𝔽7) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 0 | 4 | 0 |
| 0 | 0 | 2 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(7))| [0,0,1,1,0,0,0,1,0],[2,0,0,0,2,0,0,0,2],[0,0,1,4,0,0,0,2,0] >;
He3 in GAP, Magma, Sage, TeX
{\rm He}_3 % in TeX
G:=Group("He3"); // GroupNames label
G:=SmallGroup(27,3);
// by ID
G=gap.SmallGroup(27,3);
# by ID
G:=PCGroup([3,-3,3,-3,73]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^3=1,a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of He3 in TeX
Character table of He3 in TeX