non-abelian, supersoluble, monomial
Aliases: He3.4C6, (C3×C9)⋊8S3, C9○He3⋊2C2, C9.2(C3⋊S3), C32.5(C3×S3), He3⋊C2.2C3, C3.7(C3×C3⋊S3), SmallGroup(162,44)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3.4C6 |
Generators and relations for He3.4C6
G = < a,b,c,d | a3=b3=c3=1, d6=b, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Character table of He3.4C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | 9M | 9N | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 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 | 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 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ9 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ10 | 2 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ11 | 2 | 0 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | ζ65 | ζ65 | -1+√-3 | -1-√-3 | ζ6 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ12 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | ζ65 | -1-√-3 | ζ6 | ζ6 | ζ6 | ζ65 | -1+√-3 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ13 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | ζ6 | -1+√-3 | ζ65 | ζ65 | ζ65 | ζ6 | -1-√-3 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ14 | 2 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1-√-3 | ζ65 | ζ65 | -1+√-3 | ζ65 | ζ6 | ζ6 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ15 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1-√-3 | ζ6 | ζ6 | ζ65 | ζ65 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ16 | 2 | 0 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | ζ6 | ζ6 | -1-√-3 | -1+√-3 | ζ65 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ17 | 2 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1+√-3 | ζ6 | ζ6 | -1-√-3 | ζ6 | ζ65 | ζ65 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ18 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1+√-3 | ζ65 | ζ65 | ζ6 | ζ6 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ19 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 3ζ92 | 3ζ94 | 3ζ95 | 3ζ97 | 3ζ9 | 3ζ98 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ95 | complex faithful |
| ρ20 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 3ζ95 | 3ζ9 | 3ζ98 | 3ζ94 | 3ζ97 | 3ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | ζ98 | complex faithful |
| ρ21 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 3ζ97 | 3ζ95 | 3ζ94 | 3ζ92 | 3ζ98 | 3ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ94 | complex faithful |
| ρ22 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 3ζ97 | 3ζ95 | 3ζ94 | 3ζ92 | 3ζ98 | 3ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | ζ94 | complex faithful |
| ρ23 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 3ζ95 | 3ζ9 | 3ζ98 | 3ζ94 | 3ζ97 | 3ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ98 | complex faithful |
| ρ24 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 3ζ98 | 3ζ97 | 3ζ92 | 3ζ9 | 3ζ94 | 3ζ95 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | ζ92 | complex faithful |
| ρ25 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 3ζ9 | 3ζ92 | 3ζ97 | 3ζ98 | 3ζ95 | 3ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | ζ97 | complex faithful |
| ρ26 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 3ζ98 | 3ζ97 | 3ζ92 | 3ζ9 | 3ζ94 | 3ζ95 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ92 | complex faithful |
| ρ27 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 3ζ9 | 3ζ92 | 3ζ97 | 3ζ98 | 3ζ95 | 3ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ97 | complex faithful |
| ρ28 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 3ζ94 | 3ζ98 | 3ζ9 | 3ζ95 | 3ζ92 | 3ζ97 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | ζ9 | complex faithful |
| ρ29 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 3ζ92 | 3ζ94 | 3ζ95 | 3ζ97 | 3ζ9 | 3ζ98 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | ζ95 | complex faithful |
| ρ30 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 3ζ94 | 3ζ98 | 3ζ9 | 3ζ95 | 3ζ92 | 3ζ97 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ9 | complex faithful |
►On 27 points - transitive group 27T39
Generators in S27
(1 25 16)(2 17 26)(3 27 18)(4 19 10)(5 11 20)(6 21 12)(7 13 22)(8 23 14)(9 15 24)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(10 22 16)(11 17 23)(12 24 18)(13 19 25)(14 26 20)(15 21 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)
G:=sub<Sym(27)| (1,25,16)(2,17,26)(3,27,18)(4,19,10)(5,11,20)(6,21,12)(7,13,22)(8,23,14)(9,15,24), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (10,22,16)(11,17,23)(12,24,18)(13,19,25)(14,26,20)(15,21,27), (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)>;
G:=Group( (1,25,16)(2,17,26)(3,27,18)(4,19,10)(5,11,20)(6,21,12)(7,13,22)(8,23,14)(9,15,24), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (10,22,16)(11,17,23)(12,24,18)(13,19,25)(14,26,20)(15,21,27), (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) );
G=PermutationGroup([[(1,25,16),(2,17,26),(3,27,18),(4,19,10),(5,11,20),(6,21,12),(7,13,22),(8,23,14),(9,15,24)], [(1,7,4),(2,8,5),(3,9,6),(10,16,22),(11,17,23),(12,18,24),(13,19,25),(14,20,26),(15,21,27)], [(10,22,16),(11,17,23),(12,24,18),(13,19,25),(14,26,20),(15,21,27)], [(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)]])
G:=TransitiveGroup(27,39);
He3.4C6 is a maximal subgroup of
He3.3C12 He3.6D6 He3.C18 He3.2C18 C3≀S3⋊3C3 C3≀C3.C6 He3.5C18 3- 1+4⋊2C2 C9○He3⋊4S3
He3.4C6 is a maximal quotient of
He3.5C12 C92⋊4S3 C9×He3⋊C2 (C32×C9)⋊8S3 C9⋊C9⋊2S3 C92⋊6S3 C92⋊5S3 C9○He3⋊4S3
Matrix representation of He3.4C6 ►in GL3(𝔽19) generated by
| 1 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 7 |
| 11 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 6 | 0 | 0 |
| 0 | 0 | 6 |
| 0 | 6 | 0 |
G:=sub<GL(3,GF(19))| [1,0,0,0,11,0,0,0,7],[11,0,0,0,11,0,0,0,11],[0,0,1,1,0,0,0,1,0],[6,0,0,0,0,6,0,6,0] >;
He3.4C6 in GAP, Magma, Sage, TeX
{\rm He}_3._4C_6 % in TeX
G:=Group("He3.4C6"); // GroupNames label
G:=SmallGroup(162,44);
// by ID
G=gap.SmallGroup(162,44);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,276,182,723,253]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^6=b,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of He3.4C6 in TeX
Character table of He3.4C6 in TeX