p-group, metabelian, nilpotent (class 3), monomial
Aliases: He3⋊2C3, C3.4He3, C32.3C32, (C3×C9)⋊3C3, 3-Sylow(PSL(3,19)), SmallGroup(81,9)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for He3⋊C3
G = < a,b,c,d | a3=b3=c3=d3=1, ab=ba, cac-1=dad-1=ab-1, bc=cb, bd=db, dcd-1=abc >
Character table of He3⋊C3
| class | 1 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 9A | 9B | 9C | 9D | 9E | 9F | |
| size | 1 | 1 | 1 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ10 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ11 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ12 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98+2ζ95 | ζ97+2ζ9 | 2ζ98+ζ92 | ζ95+2ζ92 | 2ζ94+ζ9 | 2ζ97+ζ94 | complex faithful |
| ρ13 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ97+2ζ9 | ζ95+2ζ92 | 2ζ97+ζ94 | 2ζ94+ζ9 | 2ζ98+ζ92 | ζ98+2ζ95 | complex faithful |
| ρ14 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ92 | 2ζ94+ζ9 | ζ98+2ζ95 | 2ζ98+ζ92 | 2ζ97+ζ94 | ζ97+2ζ9 | complex faithful |
| ρ15 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98+ζ92 | 2ζ97+ζ94 | ζ95+2ζ92 | ζ98+2ζ95 | ζ97+2ζ9 | 2ζ94+ζ9 | complex faithful |
| ρ16 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ94+ζ9 | 2ζ98+ζ92 | ζ97+2ζ9 | 2ζ97+ζ94 | ζ98+2ζ95 | ζ95+2ζ92 | complex faithful |
| ρ17 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ97+ζ94 | ζ98+2ζ95 | 2ζ94+ζ9 | ζ97+2ζ9 | ζ95+2ζ92 | 2ζ98+ζ92 | complex faithful |
►On 27 points - transitive group 27T23
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 14 10)(2 15 11)(3 13 12)(4 27 8)(5 25 9)(6 26 7)(16 23 19)(17 24 20)(18 22 21)
(2 15 11)(3 12 13)(4 9 7)(5 6 27)(8 25 26)(16 18 20)(17 23 22)(19 21 24)
(1 5 20)(2 26 22)(3 8 16)(4 23 13)(6 18 11)(7 21 15)(9 24 10)(12 27 19)(14 25 17)
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), (1,14,10)(2,15,11)(3,13,12)(4,27,8)(5,25,9)(6,26,7)(16,23,19)(17,24,20)(18,22,21), (2,15,11)(3,12,13)(4,9,7)(5,6,27)(8,25,26)(16,18,20)(17,23,22)(19,21,24), (1,5,20)(2,26,22)(3,8,16)(4,23,13)(6,18,11)(7,21,15)(9,24,10)(12,27,19)(14,25,17)>;
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), (1,14,10)(2,15,11)(3,13,12)(4,27,8)(5,25,9)(6,26,7)(16,23,19)(17,24,20)(18,22,21), (2,15,11)(3,12,13)(4,9,7)(5,6,27)(8,25,26)(16,18,20)(17,23,22)(19,21,24), (1,5,20)(2,26,22)(3,8,16)(4,23,13)(6,18,11)(7,21,15)(9,24,10)(12,27,19)(14,25,17) );
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)], [(1,14,10),(2,15,11),(3,13,12),(4,27,8),(5,25,9),(6,26,7),(16,23,19),(17,24,20),(18,22,21)], [(2,15,11),(3,12,13),(4,9,7),(5,6,27),(8,25,26),(16,18,20),(17,23,22),(19,21,24)], [(1,5,20),(2,26,22),(3,8,16),(4,23,13),(6,18,11),(7,21,15),(9,24,10),(12,27,19),(14,25,17)]])
G:=TransitiveGroup(27,23);
►On 27 points - transitive group 27T24
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)(19 21 20)(22 24 23)(25 27 26)
(1 25 14)(2 26 13)(3 27 15)(4 21 17)(5 19 16)(6 20 18)(7 24 11)(8 22 10)(9 23 12)
(1 24 17)(2 22 16)(3 23 18)(4 27 11)(5 25 10)(6 26 12)(7 20 15)(8 21 14)(9 19 13)
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26), (1,25,14)(2,26,13)(3,27,15)(4,21,17)(5,19,16)(6,20,18)(7,24,11)(8,22,10)(9,23,12), (1,24,17)(2,22,16)(3,23,18)(4,27,11)(5,25,10)(6,26,12)(7,20,15)(8,21,14)(9,19,13)>;
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26), (1,25,14)(2,26,13)(3,27,15)(4,21,17)(5,19,16)(6,20,18)(7,24,11)(8,22,10)(9,23,12), (1,24,17)(2,22,16)(3,23,18)(4,27,11)(5,25,10)(6,26,12)(7,20,15)(8,21,14)(9,19,13) );
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,3,2),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,17,18),(19,21,20),(22,24,23),(25,27,26)], [(1,25,14),(2,26,13),(3,27,15),(4,21,17),(5,19,16),(6,20,18),(7,24,11),(8,22,10),(9,23,12)], [(1,24,17),(2,22,16),(3,23,18),(4,27,11),(5,25,10),(6,26,12),(7,20,15),(8,21,14),(9,19,13)]])
G:=TransitiveGroup(27,24);
He3⋊C3 is a maximal subgroup of
He3.2C6 He3.2S3 He3⋊S3 C92⋊C3 C92⋊2C3 C32.He3 C32.6He3 C9.He3 He3⋊C32 C9.2He3 C62.14C32 He3⋊A4
He3⋊C3 is a maximal quotient of
C32.24He3 C32.27He3 C32.29He3 C32.20He3 He3⋊C9 C92⋊C3 C92⋊2C3 C92.C3 C32.He3 C32.5He3 C32.6He3 C62.14C32 He3⋊A4
Matrix representation of He3⋊C3 ►in GL3(𝔽19) generated by
| 1 | 6 | 0 |
| 0 | 18 | 1 |
| 0 | 18 | 0 |
| 7 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 6 | 4 | 9 |
| 10 | 13 | 9 |
| 6 | 13 | 0 |
| 7 | 0 | 4 |
| 18 | 0 | 12 |
| 7 | 11 | 12 |
G:=sub<GL(3,GF(19))| [1,0,0,6,18,18,0,1,0],[7,0,0,0,7,0,0,0,7],[6,10,6,4,13,13,9,9,0],[7,18,7,0,0,11,4,12,12] >;
He3⋊C3 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes C_3 % in TeX
G:=Group("He3:C3"); // GroupNames label
G:=SmallGroup(81,9);
// by ID
G=gap.SmallGroup(81,9);
# by ID
G:=PCGroup([4,-3,3,-3,-3,97,149,434]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^3=1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a*b*c>;
// generators/relations
Export
Subgroup lattice of He3⋊C3 in TeX
Character table of He3⋊C3 in TeX