Aliases: ASL2(𝔽3), PU3(𝔽2), PSU3(𝔽2)⋊C3, C32⋊SL2(𝔽3), C3⋊S3.A4, Hessian group, SmallGroup(216,153)
Series: Derived ►Chief ►Lower central ►Upper central
| PSU3(𝔽2) — ASL2(𝔽3) |
Generators and relations for ASL2(𝔽3)
G = < a,b,c,d,e | a3=b3=c4=e3=1, d2=c2, eae-1=dbd-1=ab=ba, cac-1=b-1, dad-1=a-1b, cbc-1=a, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Character table of ASL2(𝔽3)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | |
| size | 1 | 9 | 8 | 12 | 12 | 24 | 24 | 54 | 36 | 36 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 2 | -2 | 2 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ5 | 2 | -2 | 2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ6 | 2 | -2 | 2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ7 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ8 | 8 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | orthogonal faithful |
| ρ9 | 8 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 0 | complex faithful |
| ρ10 | 8 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 0 | complex faithful |
►On 9 points: primitive, doubly transitive - transitive group 9T23
Generators in S9
(1 8 6)(2 7 3)(4 5 9)
(1 9 7)(2 6 5)(3 8 4)
(2 3 4 5)(6 7 8 9)
(2 7 4 9)(3 6 5 8)
(2 6 5)(3 4 8)
G:=sub<Sym(9)| (1,8,6)(2,7,3)(4,5,9), (1,9,7)(2,6,5)(3,8,4), (2,3,4,5)(6,7,8,9), (2,7,4,9)(3,6,5,8), (2,6,5)(3,4,8)>;
G:=Group( (1,8,6)(2,7,3)(4,5,9), (1,9,7)(2,6,5)(3,8,4), (2,3,4,5)(6,7,8,9), (2,7,4,9)(3,6,5,8), (2,6,5)(3,4,8) );
G=PermutationGroup([[(1,8,6),(2,7,3),(4,5,9)], [(1,9,7),(2,6,5),(3,8,4)], [(2,3,4,5),(6,7,8,9)], [(2,7,4,9),(3,6,5,8)], [(2,6,5),(3,4,8)]])
G:=TransitiveGroup(9,23);
►On 12 points - transitive group 12T122
Generators in S12
(1 8 6)(2 5 7)(3 10 12)
(1 6 8)(2 5 7)(4 11 9)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(5 9 7 11)(6 12 8 10)
(1 4 2)(5 6 11)(7 8 9)
G:=sub<Sym(12)| (1,8,6)(2,5,7)(3,10,12), (1,6,8)(2,5,7)(4,11,9), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,9,7,11)(6,12,8,10), (1,4,2)(5,6,11)(7,8,9)>;
G:=Group( (1,8,6)(2,5,7)(3,10,12), (1,6,8)(2,5,7)(4,11,9), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,9,7,11)(6,12,8,10), (1,4,2)(5,6,11)(7,8,9) );
G=PermutationGroup([[(1,8,6),(2,5,7),(3,10,12)], [(1,6,8),(2,5,7),(4,11,9)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,3),(2,4),(5,9,7,11),(6,12,8,10)], [(1,4,2),(5,6,11),(7,8,9)]])
G:=TransitiveGroup(12,122);
►On 24 points - transitive group 24T562
Generators in S24
(1 14 19)(3 17 16)(5 21 10)(6 11 22)(7 12 23)(8 24 9)
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 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)
(1 22 3 24)(2 21 4 23)(5 18 7 20)(6 17 8 19)(9 14 11 16)(10 13 12 15)
(2 21 22)(4 23 24)(5 6 20)(7 8 18)(9 13 12)(10 11 15)
G:=sub<Sym(24)| (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (2,21,22)(4,23,24)(5,6,20)(7,8,18)(9,13,12)(10,11,15)>;
G:=Group( (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (2,21,22)(4,23,24)(5,6,20)(7,8,18)(9,13,12)(10,11,15) );
G=PermutationGroup([[(1,14,19),(3,17,16),(5,21,10),(6,11,22),(7,12,23),(8,24,9)], [(2,15,20),(4,18,13),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,22,3,24),(2,21,4,23),(5,18,7,20),(6,17,8,19),(9,14,11,16),(10,13,12,15)], [(2,21,22),(4,23,24),(5,6,20),(7,8,18),(9,13,12),(10,11,15)]])
G:=TransitiveGroup(24,562);
►On 24 points - transitive group 24T569
Generators in S24
(1 14 19)(3 17 16)(5 21 10)(6 11 22)(7 12 23)(8 24 9)
(2 15 20)(4 18 13)(5 21 10)(6 22 11)(7 12 23)(8 9 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)
(1 22 3 24)(2 21 4 23)(5 18 7 20)(6 17 8 19)(9 14 11 16)(10 13 12 15)
(1 19 14)(2 5 11)(3 17 16)(4 7 9)(6 15 21)(8 13 23)(10 22 20)(12 24 18)
G:=sub<Sym(24)| (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (1,19,14)(2,5,11)(3,17,16)(4,7,9)(6,15,21)(8,13,23)(10,22,20)(12,24,18)>;
G:=Group( (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (2,15,20)(4,18,13)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (1,19,14)(2,5,11)(3,17,16)(4,7,9)(6,15,21)(8,13,23)(10,22,20)(12,24,18) );
G=PermutationGroup([[(1,14,19),(3,17,16),(5,21,10),(6,11,22),(7,12,23),(8,24,9)], [(2,15,20),(4,18,13),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,22,3,24),(2,21,4,23),(5,18,7,20),(6,17,8,19),(9,14,11,16),(10,13,12,15)], [(1,19,14),(2,5,11),(3,17,16),(4,7,9),(6,15,21),(8,13,23),(10,22,20),(12,24,18)]])
G:=TransitiveGroup(24,569);
►On 27 points - transitive group 27T82
Generators in S27
(1 19 17)(2 21 23)(3 11 9)(4 7 10)(5 6 8)(12 16 15)(13 18 14)(20 25 24)(22 26 27)
(1 16 18)(2 22 20)(3 8 10)(4 11 5)(6 7 9)(12 13 17)(14 19 15)(21 26 25)(23 27 24)
(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 9 6 11)(5 8 7 10)(12 19 14 17)(13 18 15 16)(20 26 22 24)(21 25 23 27)
(1 3 2)(4 26 19)(5 21 14)(6 24 17)(7 23 12)(8 22 16)(9 27 13)(10 20 18)(11 25 15)
G:=sub<Sym(27)| (1,19,17)(2,21,23)(3,11,9)(4,7,10)(5,6,8)(12,16,15)(13,18,14)(20,25,24)(22,26,27), (1,16,18)(2,22,20)(3,8,10)(4,11,5)(6,7,9)(12,13,17)(14,19,15)(21,26,25)(23,27,24), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,9,6,11)(5,8,7,10)(12,19,14,17)(13,18,15,16)(20,26,22,24)(21,25,23,27), (1,3,2)(4,26,19)(5,21,14)(6,24,17)(7,23,12)(8,22,16)(9,27,13)(10,20,18)(11,25,15)>;
G:=Group( (1,19,17)(2,21,23)(3,11,9)(4,7,10)(5,6,8)(12,16,15)(13,18,14)(20,25,24)(22,26,27), (1,16,18)(2,22,20)(3,8,10)(4,11,5)(6,7,9)(12,13,17)(14,19,15)(21,26,25)(23,27,24), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,9,6,11)(5,8,7,10)(12,19,14,17)(13,18,15,16)(20,26,22,24)(21,25,23,27), (1,3,2)(4,26,19)(5,21,14)(6,24,17)(7,23,12)(8,22,16)(9,27,13)(10,20,18)(11,25,15) );
G=PermutationGroup([[(1,19,17),(2,21,23),(3,11,9),(4,7,10),(5,6,8),(12,16,15),(13,18,14),(20,25,24),(22,26,27)], [(1,16,18),(2,22,20),(3,8,10),(4,11,5),(6,7,9),(12,13,17),(14,19,15),(21,26,25),(23,27,24)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,9,6,11),(5,8,7,10),(12,19,14,17),(13,18,15,16),(20,26,22,24),(21,25,23,27)], [(1,3,2),(4,26,19),(5,21,14),(6,24,17),(7,23,12),(8,22,16),(9,27,13),(10,20,18),(11,25,15)]])
G:=TransitiveGroup(27,82);
ASL2(𝔽3) is a maximal subgroup of
AGL2(𝔽3)
| action | f(x) | Disc(f) |
|---|---|---|
| 9T23 | x9+2x8-66x7-196x6+1022x5+3614x4-2711x3-14194x2-4931x+5714 | 28·36·52·72·716·2774·5872·19932 |
| 12T122 | x12+3x11-85x10-190x9+2069x8+3233x7-15840x6-16814x5+35725x4+31758x3-5433x2-1468x+80 | 24·716·1092·2776·45134336168774164332 |
Matrix representation of ASL2(𝔽3) ►in GL8(ℤ)
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,1,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0],[-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1],[1,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
ASL2(𝔽3) in GAP, Magma, Sage, TeX
{\rm ASL}_2({\mathbb F}_3) % in TeX
G:=Group("ASL(2,3)"); // GroupNames label
G:=SmallGroup(216,153);
// by ID
G=gap.SmallGroup(216,153);
# by ID
G:=PCGroup([6,-3,-2,2,-2,-3,3,217,55,164,116,50,4324,1210,736,142,6053,1163,161,455]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=e^3=1,d^2=c^2,e*a*e^-1=d*b*d^-1=a*b=b*a,c*a*c^-1=b^-1,d*a*d^-1=a^-1*b,c*b*c^-1=a,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations
Export
Subgroup lattice of ASL2(𝔽3) in TeX
Character table of ASL2(𝔽3) in TeX