non-abelian, soluble, monomial, rational
Aliases: PSU3(𝔽2), C32⋊Q8, C32⋊C4.2C2, C3⋊S3.2C22, Mathieu group M9, SmallGroup(72,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for PSU3(𝔽2)
G = < a,b,c,d | a3=b3=c4=1, d2=c2, dbd-1=ab=ba, cac-1=b-1, dad-1=a-1b, cbc-1=a, dcd-1=c-1 >
Character table of PSU3(𝔽2)
| class | 1 | 2 | 3 | 4A | 4B | 4C | |
| size | 1 | 9 | 8 | 18 | 18 | 18 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 2 | -2 | 2 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ6 | 8 | 0 | -1 | 0 | 0 | 0 | orthogonal faithful |
►On 9 points: primitive, sharply doubly transitive - transitive group 9T14
Generators in S9
(1 3 5)(2 6 9)(4 7 8)
(1 4 2)(3 7 6)(5 8 9)
(2 3 4 5)(6 7 8 9)
(2 7 4 9)(3 6 5 8)
G:=sub<Sym(9)| (1,3,5)(2,6,9)(4,7,8), (1,4,2)(3,7,6)(5,8,9), (2,3,4,5)(6,7,8,9), (2,7,4,9)(3,6,5,8)>;
G:=Group( (1,3,5)(2,6,9)(4,7,8), (1,4,2)(3,7,6)(5,8,9), (2,3,4,5)(6,7,8,9), (2,7,4,9)(3,6,5,8) );
G=PermutationGroup([[(1,3,5),(2,6,9),(4,7,8)], [(1,4,2),(3,7,6),(5,8,9)], [(2,3,4,5),(6,7,8,9)], [(2,7,4,9),(3,6,5,8)]])
G:=TransitiveGroup(9,14);
►On 12 points - transitive group 12T47
Generators in S12
(1 11 9)(2 10 12)(4 6 8)
(1 11 9)(2 12 10)(3 7 5)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(5 9 7 11)(6 12 8 10)
G:=sub<Sym(12)| (1,11,9)(2,10,12)(4,6,8), (1,11,9)(2,12,10)(3,7,5), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,9,7,11)(6,12,8,10)>;
G:=Group( (1,11,9)(2,10,12)(4,6,8), (1,11,9)(2,12,10)(3,7,5), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,9,7,11)(6,12,8,10) );
G=PermutationGroup([[(1,11,9),(2,10,12),(4,6,8)], [(1,11,9),(2,12,10),(3,7,5)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,3),(2,4),(5,9,7,11),(6,12,8,10)]])
G:=TransitiveGroup(12,47);
►On 18 points - transitive group 18T35
Generators in S18
(1 13 11)(2 9 7)(3 12 4)(5 6 14)(8 17 16)(10 18 15)
(1 14 12)(2 10 8)(3 11 6)(4 13 5)(7 15 16)(9 18 17)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 8 5 10)(4 7 6 9)(11 15 13 17)(12 18 14 16)
G:=sub<Sym(18)| (1,13,11)(2,9,7)(3,12,4)(5,6,14)(8,17,16)(10,18,15), (1,14,12)(2,10,8)(3,11,6)(4,13,5)(7,15,16)(9,18,17), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,8,5,10)(4,7,6,9)(11,15,13,17)(12,18,14,16)>;
G:=Group( (1,13,11)(2,9,7)(3,12,4)(5,6,14)(8,17,16)(10,18,15), (1,14,12)(2,10,8)(3,11,6)(4,13,5)(7,15,16)(9,18,17), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,8,5,10)(4,7,6,9)(11,15,13,17)(12,18,14,16) );
G=PermutationGroup([[(1,13,11),(2,9,7),(3,12,4),(5,6,14),(8,17,16),(10,18,15)], [(1,14,12),(2,10,8),(3,11,6),(4,13,5),(7,15,16),(9,18,17)], [(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(1,2),(3,8,5,10),(4,7,6,9),(11,15,13,17),(12,18,14,16)]])
G:=TransitiveGroup(18,35);
►On 24 points - transitive group 24T82
Generators in S24
(2 20 15)(4 13 18)(5 10 21)(6 11 22)(7 23 12)(8 24 9)
(1 14 19)(3 17 16)(5 21 10)(6 11 22)(7 12 23)(8 24 9)
(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
G:=sub<Sym(24)| (2,20,15)(4,13,18)(5,10,21)(6,11,22)(7,23,12)(8,24,9), (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;
G:=Group( (2,20,15)(4,13,18)(5,10,21)(6,11,22)(7,23,12)(8,24,9), (1,14,19)(3,17,16)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );
G=PermutationGroup([[(2,20,15),(4,13,18),(5,10,21),(6,11,22),(7,23,12),(8,24,9)], [(1,14,19),(3,17,16),(5,21,10),(6,11,22),(7,12,23),(8,24,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)]])
G:=TransitiveGroup(24,82);
PSU3(𝔽2) is a maximal subgroup of
AΓL1(𝔽9) ASL2(𝔽3) C33⋊Q8 C32⋊Dic10
PSU3(𝔽2) is a maximal quotient of C2.PSU3(𝔽2) SU3(𝔽2) C33⋊Q8 C32⋊Dic10
| action | f(x) | Disc(f) |
|---|---|---|
| 9T14 | x9-2x8-60x7+120x6+980x5-1808x4-4012x3+4936x2+4673x-1434 | 224·34·76·116·715689152892 |
| 12T47 | x12-162x10-480x9+6213x8+31488x7+11624x6-74304x5+28917x4+10080x3-4602x2+49 | 250·312·72·192·476·592·3372·13072·28972·1118632 |
Matrix representation of PSU3(𝔽2) ►in GL8(ℤ)
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 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 |
G:=sub<GL(8,Integers())| [0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0],[0,0,1,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,1,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,1,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,0],[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] >;
PSU3(𝔽2) in GAP, Magma, Sage, TeX
{\rm PSU}_3({\mathbb F}_2) % in TeX
G:=Group("PSU(3,2)"); // GroupNames label
G:=SmallGroup(72,41);
// by ID
G=gap.SmallGroup(72,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,3,20,61,26,1123,248,93,1604,209,314]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=1,d^2=c^2,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,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of PSU3(𝔽2) in TeX
Character table of PSU3(𝔽2) in TeX