p-group, metabelian, nilpotent (class 3), monomial
Aliases: C3.5He3, C32.4C32, 3- 1+2.C3, (C3×C9).2C3, SmallGroup(81,10)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C3.He3
G = < a,b,c,d | a3=c3=1, b3=a-1, d3=a, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=abc-1, dcd-1=a-1c >
Character table of C3.He3
| class | 1 | 3A | 3B | 3C | 3D | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 9J | 9K | 9L | |
| size | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | |
| 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 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | 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 | ζ97+2ζ9 | 2ζ94+ζ9 | ζ98+2ζ95 | 2ζ98+ζ92 | 2ζ97+ζ94 | ζ95+2ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ13 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ98+2ζ95 | ζ95+2ζ92 | 2ζ97+ζ94 | 2ζ94+ζ9 | 2ζ98+ζ92 | ζ97+2ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ14 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ95+2ζ92 | 2ζ98+ζ92 | ζ97+2ζ9 | 2ζ97+ζ94 | ζ98+2ζ95 | 2ζ94+ζ9 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 2ζ94+ζ9 | 2ζ97+ζ94 | ζ95+2ζ92 | ζ98+2ζ95 | ζ97+2ζ9 | 2ζ98+ζ92 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 2ζ97+ζ94 | ζ97+2ζ9 | 2ζ98+ζ92 | ζ95+2ζ92 | 2ζ94+ζ9 | ζ98+2ζ95 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ17 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 2ζ98+ζ92 | ζ98+2ζ95 | 2ζ94+ζ9 | ζ97+2ζ9 | ζ95+2ζ92 | 2ζ97+ζ94 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 27 points - transitive group 27T25
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)
(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)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)
(1 26 11 7 23 17 4 20 14)(2 21 12 8 27 18 5 24 15)(3 25 13 9 22 10 6 19 16)
G:=sub<Sym(27)| (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (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), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (1,26,11,7,23,17,4,20,14)(2,21,12,8,27,18,5,24,15)(3,25,13,9,22,10,6,19,16)>;
G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (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), (10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27), (1,26,11,7,23,17,4,20,14)(2,21,12,8,27,18,5,24,15)(3,25,13,9,22,10,6,19,16) );
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24)], [(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)], [(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27)], [(1,26,11,7,23,17,4,20,14),(2,21,12,8,27,18,5,24,15),(3,25,13,9,22,10,6,19,16)]])
G:=TransitiveGroup(27,25);
C3.He3 is a maximal subgroup of
3- 1+2.S3 C92⋊C3 C92.C3 C32.He3 C32.5He3 C32.6He3 C9.He3 C32.C33 C9.2He3 C62.15C32 C62.C32
C3.He3 is a maximal quotient of
C33.3C32 C32.28He3 C32.29He3 C32.20He3 3- 1+2⋊C9 C62.15C32 C62.C32
Matrix representation of C3.He3 ►in GL3(𝔽19) generated by
| 11 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 16 | 0 | 6 |
| 1 | 0 | 0 |
| 6 | 11 | 0 |
| 10 | 0 | 7 |
| 6 | 10 | 0 |
| 12 | 13 | 1 |
| 12 | 15 | 0 |
G:=sub<GL(3,GF(19))| [11,0,0,0,11,0,0,0,11],[4,0,16,0,4,0,0,0,6],[1,6,10,0,11,0,0,0,7],[6,12,12,10,13,15,0,1,0] >;
C3.He3 in GAP, Magma, Sage, TeX
C_3.{\rm He}_3 % in TeX
G:=Group("C3.He3"); // GroupNames label
G:=SmallGroup(81,10);
// by ID
G=gap.SmallGroup(81,10);
# by ID
G:=PCGroup([4,-3,3,-3,-3,108,97,149,434]);
// Polycyclic
G:=Group<a,b,c,d|a^3=c^3=1,b^3=a^-1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a*b*c^-1,d*c*d^-1=a^-1*c>;
// generators/relations
Export
Subgroup lattice of C3.He3 in TeX
Character table of C3.He3 in TeX