Aliases: U2(𝔽3), C4.5S4, Q8.Dic3, SL2(𝔽3)⋊2C4, C4○D4.1S3, C4.A4.2C2, C2.3(A4⋊C4), SmallGroup(96,67)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — U2(𝔽3) |
Generators and relations for U2(𝔽3)
G = < a,b,c,d,e | a4=d3=1, b2=c2=a2, e2=a, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=a2b, dbd-1=a2bc, ebe-1=bc, dcd-1=b, ece-1=a2c, ede-1=d-1 >
Character table of U2(𝔽3)
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6 | 8A | 8B | 12A | 12B | |
| size | 1 | 1 | 6 | 8 | 1 | 1 | 6 | 6 | 6 | 6 | 6 | 8 | 12 | 12 | 8 | 8 | |
| 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 | -i | i | i | -i | 1 | 1 | -i | i | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | -1 | i | -i | -i | i | 1 | 1 | i | -i | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 2 | -2 | 0 | -1 | 2i | -2i | -1+i | 1+i | -1-i | 1-i | 0 | 1 | 0 | 0 | -i | i | complex faithful |
| ρ8 | 2 | -2 | 0 | -1 | -2i | 2i | 1+i | -1+i | 1-i | -1-i | 0 | 1 | 0 | 0 | i | -i | complex faithful |
| ρ9 | 2 | -2 | 0 | -1 | 2i | -2i | 1-i | -1-i | 1+i | -1+i | 0 | 1 | 0 | 0 | -i | i | complex faithful |
| ρ10 | 2 | -2 | 0 | -1 | -2i | 2i | -1-i | 1-i | -1+i | 1+i | 0 | 1 | 0 | 0 | i | -i | complex faithful |
| ρ11 | 3 | 3 | -1 | 0 | 3 | 3 | 1 | 1 | 1 | 1 | -1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ12 | 3 | 3 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S4 |
| ρ13 | 3 | 3 | 1 | 0 | -3 | -3 | i | -i | -i | i | -1 | 0 | -i | i | 0 | 0 | complex lifted from A4⋊C4 |
| ρ14 | 3 | 3 | 1 | 0 | -3 | -3 | -i | i | i | -i | -1 | 0 | i | -i | 0 | 0 | complex lifted from A4⋊C4 |
| ρ15 | 4 | -4 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | i | -i | complex faithful |
| ρ16 | 4 | -4 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -i | i | complex faithful |
►On 24 points - transitive group 24T138
Generators in S24
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)
(1 4 5 8)(2 3 6 7)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 20 22 24)
(1 3 5 7)(2 8 6 4)(9 23 13 19)(10 20 14 24)(11 17 15 21)(12 22 16 18)
(1 15 18)(2 19 16)(3 9 20)(4 21 10)(5 11 22)(6 23 12)(7 13 24)(8 17 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24), (1,4,5,8)(2,3,6,7)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,20,22,24), (1,3,5,7)(2,8,6,4)(9,23,13,19)(10,20,14,24)(11,17,15,21)(12,22,16,18), (1,15,18)(2,19,16)(3,9,20)(4,21,10)(5,11,22)(6,23,12)(7,13,24)(8,17,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24), (1,4,5,8)(2,3,6,7)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,20,22,24), (1,3,5,7)(2,8,6,4)(9,23,13,19)(10,20,14,24)(11,17,15,21)(12,22,16,18), (1,15,18)(2,19,16)(3,9,20)(4,21,10)(5,11,22)(6,23,12)(7,13,24)(8,17,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24)], [(1,4,5,8),(2,3,6,7),(9,17,13,21),(10,16,14,12),(11,19,15,23),(18,20,22,24)], [(1,3,5,7),(2,8,6,4),(9,23,13,19),(10,20,14,24),(11,17,15,21),(12,22,16,18)], [(1,15,18),(2,19,16),(3,9,20),(4,21,10),(5,11,22),(6,23,12),(7,13,24),(8,17,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,138);
U2(𝔽3) is a maximal subgroup of
CU2(𝔽3) C8.5S4 U2(𝔽3)⋊C2 Q8.4S4 Q8.5S4 D4.S4 D4.3S4 C3⋊U2(𝔽3) GL2(𝔽5) C5⋊2U2(𝔽3) C5⋊U2(𝔽3)
U2(𝔽3) is a maximal quotient of
C2.U2(𝔽3) C12.9S4 C3⋊U2(𝔽3) C5⋊2U2(𝔽3) C5⋊U2(𝔽3)
Matrix representation of U2(𝔽3) ►in GL2(𝔽5) generated by
| 2 | 0 |
| 0 | 2 |
| 2 | 2 |
| 0 | 3 |
| 3 | 0 |
| 4 | 2 |
| 0 | 2 |
| 2 | 4 |
| 4 | 4 |
| 4 | 1 |
G:=sub<GL(2,GF(5))| [2,0,0,2],[2,0,2,3],[3,4,0,2],[0,2,2,4],[4,4,4,1] >;
U2(𝔽3) in GAP, Magma, Sage, TeX
{\rm U}_2({\mathbb F}_3) % in TeX
G:=Group("U(2,3)"); // GroupNames label
G:=SmallGroup(96,67);
// by ID
G=gap.SmallGroup(96,67);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,12,295,146,579,447,117,364,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=d^3=1,b^2=c^2=a^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b*c,e*b*e^-1=b*c,d*c*d^-1=b,e*c*e^-1=a^2*c,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of U2(𝔽3) in TeX
Character table of U2(𝔽3) in TeX