Aliases: SU3(𝔽2), CSU3(𝔽2), He3⋊Q8, C3.PSU3(𝔽2), He3⋊C4.2C2, He3⋊C2.1C22, SmallGroup(216,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SU3(𝔽2)
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, cac-1=ab-1, dad-1=cb=bc, eae-1=ac, bd=db, be=eb, dcd-1=a-1, ece-1=ab-1c-1, ede-1=d-1 >
Character table of SU3(𝔽2)
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 9 | 1 | 1 | 24 | 18 | 18 | 18 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ6 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | -1 | 1 | -1 | ζ6 | ζ65 | ζ6 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | complex faithful |
| ρ7 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 1 | 1 | 1 | ζ65 | ζ6 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | complex faithful |
| ρ8 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | -1 | -1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ65 | ζ3 | complex faithful |
| ρ9 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | -1 | 1 | -1 | ζ65 | ζ6 | ζ65 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | complex faithful |
| ρ10 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 1 | 1 | 1 | ζ6 | ζ65 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | complex faithful |
| ρ11 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | -1 | -1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ6 | ζ32 | complex faithful |
| ρ12 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 1 | -1 | -1 | ζ65 | ζ6 | ζ3 | ζ6 | ζ65 | ζ65 | ζ32 | ζ6 | complex faithful |
| ρ13 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 1 | -1 | -1 | ζ6 | ζ65 | ζ32 | ζ65 | ζ6 | ζ6 | ζ3 | ζ65 | complex faithful |
| ρ14 | 6 | 2 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 6 | 2 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 8 | 0 | 8 | 8 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from PSU3(𝔽2) |
►On 27 points - transitive group 27T83
Generators in S27
(1 4 10)(2 8 21)(3 23 6)(5 14 17)(7 27 26)(9 24 15)(11 19 18)(12 22 13)(16 25 20)
(1 3 2)(4 23 8)(5 20 9)(6 21 10)(7 22 11)(12 18 26)(13 19 27)(14 16 24)(15 17 25)
(1 11 5)(2 22 9)(3 7 20)(4 27 24)(6 18 15)(8 19 16)(10 12 25)(13 14 23)(17 21 26)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
(4 25 6 27)(5 24 7 26)(8 17 10 19)(9 16 11 18)(12 20 14 22)(13 23 15 21)
G:=sub<Sym(27)| (1,4,10)(2,8,21)(3,23,6)(5,14,17)(7,27,26)(9,24,15)(11,19,18)(12,22,13)(16,25,20), (1,3,2)(4,23,8)(5,20,9)(6,21,10)(7,22,11)(12,18,26)(13,19,27)(14,16,24)(15,17,25), (1,11,5)(2,22,9)(3,7,20)(4,27,24)(6,18,15)(8,19,16)(10,12,25)(13,14,23)(17,21,26), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,25,6,27)(5,24,7,26)(8,17,10,19)(9,16,11,18)(12,20,14,22)(13,23,15,21)>;
G:=Group( (1,4,10)(2,8,21)(3,23,6)(5,14,17)(7,27,26)(9,24,15)(11,19,18)(12,22,13)(16,25,20), (1,3,2)(4,23,8)(5,20,9)(6,21,10)(7,22,11)(12,18,26)(13,19,27)(14,16,24)(15,17,25), (1,11,5)(2,22,9)(3,7,20)(4,27,24)(6,18,15)(8,19,16)(10,12,25)(13,14,23)(17,21,26), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,25,6,27)(5,24,7,26)(8,17,10,19)(9,16,11,18)(12,20,14,22)(13,23,15,21) );
G=PermutationGroup([[(1,4,10),(2,8,21),(3,23,6),(5,14,17),(7,27,26),(9,24,15),(11,19,18),(12,22,13),(16,25,20)], [(1,3,2),(4,23,8),(5,20,9),(6,21,10),(7,22,11),(12,18,26),(13,19,27),(14,16,24),(15,17,25)], [(1,11,5),(2,22,9),(3,7,20),(4,27,24),(6,18,15),(8,19,16),(10,12,25),(13,14,23),(17,21,26)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,25,6,27),(5,24,7,26),(8,17,10,19),(9,16,11,18),(12,20,14,22),(13,23,15,21)]])
G:=TransitiveGroup(27,83);
SU3(𝔽2) is a maximal subgroup of
He3⋊SD16
SU3(𝔽2) is a maximal quotient of C2.SU3(𝔽2)
Matrix representation of SU3(𝔽2) ►in GL3(𝔽7) generated by
| 0 | 6 | 3 |
| 0 | 2 | 0 |
| 1 | 1 | 5 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 2 | 0 | 0 |
| 3 | 0 | 3 |
| 1 | 1 | 5 |
| 6 | 2 | 3 |
| 0 | 3 | 5 |
| 0 | 5 | 4 |
| 1 | 1 | 5 |
| 0 | 5 | 4 |
| 0 | 4 | 2 |
G:=sub<GL(3,GF(7))| [0,0,1,6,2,1,3,0,5],[4,0,0,0,4,0,0,0,4],[2,3,1,0,0,1,0,3,5],[6,0,0,2,3,5,3,5,4],[1,0,0,1,5,4,5,4,2] >;
SU3(𝔽2) in GAP, Magma, Sage, TeX
{\rm SU}_3({\mathbb F}_2) % in TeX
G:=Group("SU(3,2)"); // GroupNames label
G:=SmallGroup(216,88);
// by ID
G=gap.SmallGroup(216,88);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,3,-3,24,73,31,1347,297,543,6244,1330,916,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=c*b=b*c,e*a*e^-1=a*c,b*d=d*b,b*e=e*b,d*c*d^-1=a^-1,e*c*e^-1=a*b^-1*c^-1,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of SU3(𝔽2) in TeX
Character table of SU3(𝔽2) in TeX