non-abelian, supersoluble, monomial
Aliases: He3⋊2C2, C32⋊2S3, C3.2(C3⋊S3), Aut(3- 1+2), SmallGroup(54,8)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3⋊C2 |
Generators and relations for He3⋊C2
G = < a,b,c,d | a3=b3=c3=d2=1, ab=ba, cac-1=ab-1, dad=a-1, bc=cb, bd=db, dcd=c-1 >
Character table of He3⋊C2
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | |
| size | 1 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 2 | 0 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | complex faithful |
| ρ8 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | complex faithful |
| ρ9 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | complex faithful |
| ρ10 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | complex faithful |
►On 9 points - transitive group 9T12
Generators in S9
(1 2 3)(4 5 6)(7 8 9)
(1 9 5)(2 7 6)(3 8 4)
(2 7 6)(3 4 8)
(2 3)(4 6)(7 8)
G:=sub<Sym(9)| (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4), (2,7,6)(3,4,8), (2,3)(4,6)(7,8)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9), (1,9,5)(2,7,6)(3,8,4), (2,7,6)(3,4,8), (2,3)(4,6)(7,8) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9)], [(1,9,5),(2,7,6),(3,8,4)], [(2,7,6),(3,4,8)], [(2,3),(4,6),(7,8)]])
G:=TransitiveGroup(9,12);
►On 18 points - transitive group 18T24
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 11 13)(2 12 14)(3 10 15)(4 9 18)(5 7 16)(6 8 17)
(2 12 14)(3 15 10)(4 9 18)(5 16 7)
(1 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 17)(14 16)(15 18)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11,13)(2,12,14)(3,10,15)(4,9,18)(5,7,16)(6,8,17), (2,12,14)(3,15,10)(4,9,18)(5,16,7), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,17)(14,16)(15,18)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11,13)(2,12,14)(3,10,15)(4,9,18)(5,7,16)(6,8,17), (2,12,14)(3,15,10)(4,9,18)(5,16,7), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,17)(14,16)(15,18) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,11,13),(2,12,14),(3,10,15),(4,9,18),(5,7,16),(6,8,17)], [(2,12,14),(3,15,10),(4,9,18),(5,16,7)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,17),(14,16),(15,18)]])
G:=TransitiveGroup(18,24);
►On 27 points - transitive group 27T6
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)
(2 3)(4 22)(5 24)(6 23)(7 20)(8 19)(9 21)(10 12)(14 15)(16 26)(17 25)(18 27)
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), (2,3)(4,22)(5,24)(6,23)(7,20)(8,19)(9,21)(10,12)(14,15)(16,26)(17,25)(18,27)>;
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), (2,3)(4,22)(5,24)(6,23)(7,20)(8,19)(9,21)(10,12)(14,15)(16,26)(17,25)(18,27) );
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)], [(2,3),(4,22),(5,24),(6,23),(7,20),(8,19),(9,21),(10,12),(14,15),(16,26),(17,25),(18,27)]])
G:=TransitiveGroup(27,6);
He3⋊C2 is a maximal subgroup of
He3⋊C4 C32⋊D6 C3≀S3 He3.C6 He3.2C6 He3.4C6 He3⋊5S3 C32⋊S4 C32⋊D15 C32⋊D21
He3⋊C2 is a maximal quotient of
He3⋊3C4 C32⋊2D9 C33⋊S3 He3.3S3 He3⋊S3 3- 1+2.S3 He3⋊5S3 C32⋊S4 C32⋊D15 C32⋊D21
| action | f(x) | Disc(f) |
|---|---|---|
| 9T12 | x9+x8-54x7+68x6+695x5-1857x4-473x3+6301x2-7401x+2727 | 212·325·114·472·1073 |
Matrix representation of He3⋊C2 ►in GL3(𝔽7) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 0 | 4 | 0 |
| 0 | 0 | 1 |
| 2 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(7))| [0,0,1,1,0,0,0,1,0],[4,0,0,0,4,0,0,0,4],[0,0,2,4,0,0,0,1,0],[1,0,0,0,0,1,0,1,0] >;
He3⋊C2 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes C_2 % in TeX
G:=Group("He3:C2"); // GroupNames label
G:=SmallGroup(54,8);
// by ID
G=gap.SmallGroup(54,8);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,33,146,150]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of He3⋊C2 in TeX
Character table of He3⋊C2 in TeX