Aliases: CSU2(𝔽3), Q8.S3, C2.2S4, SL2(𝔽3).C2, Binary octahedral group (2O, <2,3,4>), SmallGroup(48,28)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — CSU2(𝔽3) |
Generators and relations for CSU2(𝔽3)
G = < a,b,c,d | a4=c3=1, b2=d2=a2, bab-1=dbd-1=a-1, cac-1=ab, dad-1=a2b, cbc-1=a, dcd-1=c-1 >
Character table of CSU2(𝔽3)
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | |
| size | 1 | 1 | 8 | 6 | 12 | 8 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 2 | 2 | -1 | 2 | 0 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 2 | -2 | -1 | 0 | 0 | 1 | √2 | -√2 | symplectic faithful, Schur index 2 |
| ρ5 | 2 | -2 | -1 | 0 | 0 | 1 | -√2 | √2 | symplectic faithful, Schur index 2 |
| ρ6 | 3 | 3 | 0 | -1 | -1 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ7 | 3 | 3 | 0 | -1 | 1 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ8 | 4 | -4 | 1 | 0 | 0 | -1 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 16 points - transitive group 16T65
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5 3 7)(2 8 4 6)(9 14 11 16)(10 13 12 15)
(2 5 8)(4 7 6)(9 15 14)(11 13 16)
(1 10 3 12)(2 15 4 13)(5 9 7 11)(6 16 8 14)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5,3,7)(2,8,4,6)(9,14,11,16)(10,13,12,15), (2,5,8)(4,7,6)(9,15,14)(11,13,16), (1,10,3,12)(2,15,4,13)(5,9,7,11)(6,16,8,14)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5,3,7)(2,8,4,6)(9,14,11,16)(10,13,12,15), (2,5,8)(4,7,6)(9,15,14)(11,13,16), (1,10,3,12)(2,15,4,13)(5,9,7,11)(6,16,8,14) );
G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,5,3,7),(2,8,4,6),(9,14,11,16),(10,13,12,15)], [(2,5,8),(4,7,6),(9,15,14),(11,13,16)], [(1,10,3,12),(2,15,4,13),(5,9,7,11),(6,16,8,14)])
G:=TransitiveGroup(16,65);
CSU2(𝔽3) is a maximal subgroup of
Q8.D6 C4.S4 C4.6S4 C6.5S4 Q8.1S4 Q8.S4 CSU2(𝔽5) Q8.D15 SL2(𝔽7) Q8.D21
CSU2(𝔽3) is a maximal quotient of
Q8⋊Dic3 Q8.D9 C6.5S4 C23.7S4 Q8.1S4 Q8.D15 Q8.D21
Matrix representation of CSU2(𝔽3) ►in GL2(𝔽7) generated by
| 2 | 1 |
| 2 | 5 |
| 1 | 2 |
| 6 | 6 |
| 4 | 0 |
| 2 | 2 |
| 2 | 5 |
| 6 | 5 |
G:=sub<GL(2,GF(7))| [2,2,1,5],[1,6,2,6],[4,2,0,2],[2,6,5,5] >;
CSU2(𝔽3) in GAP, Magma, Sage, TeX
{\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("CSU(2,3)"); // GroupNames label
G:=SmallGroup(48,28);
// by ID
G=gap.SmallGroup(48,28);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-2,120,41,182,277,72,123,188,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^3=1,b^2=d^2=a^2,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=a*b,d*a*d^-1=a^2*b,c*b*c^-1=a,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of CSU2(𝔽3) in TeX
Character table of CSU2(𝔽3) in TeX