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## G = PSU3(𝔽2)  order 72 = 23·32

### Projective special unitary group on 𝔽23

Aliases: PSU3(𝔽2), C32⋊Q8, C32⋊C4.2C2, C3⋊S3.2C22, Mathieu group M9, SmallGroup(72,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — PSU3(𝔽2)
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — PSU3(𝔽2)
 Lower central C32 — C3⋊S3 — PSU3(𝔽2)
 Upper central C1

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

Permutation representations of PSU3(𝔽2)
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 7 5)(3 12 10)(4 9 11)
(2 8 6)(3 10 12)(4 9 11)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(5 12 7 10)(6 11 8 9)

G:=sub<Sym(12)| (1,7,5)(3,12,10)(4,9,11), (2,8,6)(3,10,12)(4,9,11), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,12,7,10)(6,11,8,9)>;

G:=Group( (1,7,5)(3,12,10)(4,9,11), (2,8,6)(3,10,12)(4,9,11), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,3)(2,4)(5,12,7,10)(6,11,8,9) );

G=PermutationGroup([(1,7,5),(3,12,10),(4,9,11)], [(2,8,6),(3,10,12),(4,9,11)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,3),(2,4),(5,12,7,10),(6,11,8,9)])

G:=TransitiveGroup(12,47);

On 18 points - transitive group 18T35
Generators in S18
(1 8 10)(2 4 6)(3 17 16)(5 18 15)(7 13 12)(9 14 11)
(1 9 7)(2 5 3)(4 18 17)(6 15 16)(8 14 13)(10 11 12)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 14 5 12)(4 13 6 11)(7 18 9 16)(8 17 10 15)

G:=sub<Sym(18)| (1,8,10)(2,4,6)(3,17,16)(5,18,15)(7,13,12)(9,14,11), (1,9,7)(2,5,3)(4,18,17)(6,15,16)(8,14,13)(10,11,12), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,14,5,12)(4,13,6,11)(7,18,9,16)(8,17,10,15)>;

G:=Group( (1,8,10)(2,4,6)(3,17,16)(5,18,15)(7,13,12)(9,14,11), (1,9,7)(2,5,3)(4,18,17)(6,15,16)(8,14,13)(10,11,12), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (1,2)(3,14,5,12)(4,13,6,11)(7,18,9,16)(8,17,10,15) );

G=PermutationGroup([(1,8,10),(2,4,6),(3,17,16),(5,18,15),(7,13,12),(9,14,11)], [(1,9,7),(2,5,3),(4,18,17),(6,15,16),(8,14,13),(10,11,12)], [(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(1,2),(3,14,5,12),(4,13,6,11),(7,18,9,16),(8,17,10,15)])

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

Polynomial with Galois group PSU3(𝔽2) over ℚ
actionf(x)Disc(f)
9T14x9-2x8-60x7+120x6+980x5-1808x4-4012x3+4936x2+4673x-1434224·34·76·116·715689152892
12T47x12-162x10-480x9+6213x8+31488x7+11624x6-74304x5+28917x4+10080x3-4602x2+49250·312·72·192·476·592·3372·13072·28972·1118632

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

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