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G = PSU3(𝔽2)  order 72 = 23·32

Projective special unitary group on 𝔽23

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

C1C32C3⋊S3 — PSU3(𝔽2)
C1C32C3⋊S3C32⋊C4 — PSU3(𝔽2)
C32C3⋊S3 — PSU3(𝔽2)
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 >

9C2
4C3
9C4
9C4
9C4
12S3
9Q8

Character table of PSU3(𝔽2)

 class 1234A4B4C
 size 198181818
ρ1111111    trivial
ρ2111-11-1    linear of order 2
ρ31111-1-1    linear of order 2
ρ4111-1-11    linear of order 2
ρ52-22000    symplectic lifted from Q8, Schur index 2
ρ680-1000    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)

00010000
00001000
00000100
00000010
00000001
-1-1-1-1-1-1-1-1
10000000
01000000
,
01000000
00100000
10000000
00001000
00000100
00010000
00000001
-1-1-1-1-1-1-1-1
,
10000000
00010000
00000010
00100000
00000100
-1-1-1-1-1-1-1-1
01000000
00001000
,
10000000
00001000
-1-1-1-1-1-1-1-1
00000001
00100000
00010000
00000100
00000010

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

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