p-group, metacyclic, nilpotent (class 2), monomial
Aliases: 3- 1+2, C9⋊C3, C32.C3, C3.2C32, 3-Sylow(AGammaL(1,64)), SmallGroup(27,4)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for 3- 1+2
G = < a,b | a9=b3=1, bab-1=a4 >
Character table of 3- 1+2
| class | 1 | 3A | 3B | 3C | 3D | 9A | 9B | 9C | 9D | 9E | 9F | |
| 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 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 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 9T6
Generators in S9
(1 2 3 4 5 6 7 8 9)
(2 8 5)(3 6 9)
G:=sub<Sym(9)| (1,2,3,4,5,6,7,8,9), (2,8,5)(3,6,9)>;
G:=Group( (1,2,3,4,5,6,7,8,9), (2,8,5)(3,6,9) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9)], [(2,8,5),(3,6,9)]])
G:=TransitiveGroup(9,6);
►Regular action on 27 points - transitive group 27T5
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 21 13)(2 19 17)(3 26 12)(4 24 16)(5 22 11)(6 20 15)(7 27 10)(8 25 14)(9 23 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), (1,21,13)(2,19,17)(3,26,12)(4,24,16)(5,22,11)(6,20,15)(7,27,10)(8,25,14)(9,23,18)>;
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), (1,21,13)(2,19,17)(3,26,12)(4,24,16)(5,22,11)(6,20,15)(7,27,10)(8,25,14)(9,23,18) );
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)], [(1,21,13),(2,19,17),(3,26,12),(4,24,16),(5,22,11),(6,20,15),(7,27,10),(8,25,14),(9,23,18)]])
G:=TransitiveGroup(27,5);
3- 1+2 is a maximal subgroup of
C9⋊C6 C3≀C3 He3.C3 C3.He3 C9○He3 C9⋊A4 C32.A4 C63⋊C3 C63⋊3C3 C21.C32 C117⋊C3 C117⋊3C3 C39.C32
3- 1+2 is a maximal quotient of
C32⋊C9 C9⋊C9 C9⋊A4 C32.A4 C63⋊C3 C63⋊3C3 C21.C32 C117⋊C3 C117⋊3C3 C39.C32
| action | f(x) | Disc(f) |
|---|---|---|
| 9T6 | x9-21x7-7x6+126x5+42x4-273x3-63x2+189x+7 | 312·78·2151532 |
Matrix representation of 3- 1+2 ►in GL3(𝔽7) generated by
| 0 | 0 | 1 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 2 |
G:=sub<GL(3,GF(7))| [0,4,0,0,0,4,1,0,0],[1,0,0,0,4,0,0,0,2] >;
3- 1+2 in GAP, Magma, Sage, TeX
3_-^{1+2} % in TeX
G:=Group("ES-(3,1)"); // GroupNames label
G:=SmallGroup(27,4);
// by ID
G=gap.SmallGroup(27,4);
# by ID
G:=PCGroup([3,-3,3,-3,27,73]);
// Polycyclic
G:=Group<a,b|a^9=b^3=1,b*a*b^-1=a^4>;
// generators/relations
Export
Subgroup lattice of 3- 1+2 in TeX
Character table of 3- 1+2 in TeX