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G = SU3(𝔽2)  order 216 = 23·33

Special unitary group on 𝔽23

non-abelian, soluble

Aliases: SU3(𝔽2), CSU3(𝔽2), He3⋊Q8, C3.PSU3(𝔽2), He3⋊C4.2C2, He3⋊C2.1C22, SmallGroup(216,88)

Series: Derived Chief Lower central Upper central

C1C3He3He3⋊C2 — SU3(𝔽2)
C1C3He3He3⋊C2He3⋊C4 — SU3(𝔽2)
He3He3⋊C2 — SU3(𝔽2)
C1C3

Generators and relations for SU3(𝔽2)
 G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, cac-1=ab-1, dad-1=cb=bc, eae-1=ac, bd=db, be=eb, dcd-1=a-1, ece-1=ab-1c-1, ede-1=d-1 >

9C2
12C3
9C4
9C4
9C4
9C6
12S3
4C32
9Q8
9C12
9C12
9C12
12C3×S3
9C3×Q8

Character table of SU3(𝔽2)

 class 123A3B3C4A4B4C6A6B12A12B12C12D12E12F
 size 19112418181899181818181818
ρ11111111111111111    trivial
ρ211111-1-1111-1-1-11-11    linear of order 2
ρ3111111-1-1111-1-1-11-1    linear of order 2
ρ411111-11-111-111-1-1-1    linear of order 2
ρ52-2222000-2-2000000    symplectic lifted from Q8, Schur index 2
ρ63-1-3-3-3/2-3+3-3/20-11-1ζ6ζ65ζ6ζ3ζ32ζ6ζ65ζ65    complex faithful
ρ73-1-3+3-3/2-3-3-3/20111ζ65ζ6ζ3ζ32ζ3ζ3ζ32ζ32    complex faithful
ρ83-1-3-3-3/2-3+3-3/20-1-11ζ6ζ65ζ6ζ65ζ6ζ32ζ65ζ3    complex faithful
ρ93-1-3+3-3/2-3-3-3/20-11-1ζ65ζ6ζ65ζ32ζ3ζ65ζ6ζ6    complex faithful
ρ103-1-3-3-3/2-3+3-3/20111ζ6ζ65ζ32ζ3ζ32ζ32ζ3ζ3    complex faithful
ρ113-1-3+3-3/2-3-3-3/20-1-11ζ65ζ6ζ65ζ6ζ65ζ3ζ6ζ32    complex faithful
ρ123-1-3+3-3/2-3-3-3/201-1-1ζ65ζ6ζ3ζ6ζ65ζ65ζ32ζ6    complex faithful
ρ133-1-3-3-3/2-3+3-3/201-1-1ζ6ζ65ζ32ζ65ζ6ζ6ζ3ζ65    complex faithful
ρ1462-3-3-3-3+3-30000-1--3-1+-3000000    complex faithful
ρ1562-3+3-3-3-3-30000-1+-3-1--3000000    complex faithful
ρ168088-100000000000    orthogonal lifted from PSU3(𝔽2)

Permutation representations of SU3(𝔽2)
On 27 points - transitive group 27T83
Generators in S27
(1 16 27)(2 25 9)(3 11 18)(4 22 8)(5 17 14)(6 24 7)(10 13 12)(15 26 21)(19 20 23)
(1 3 2)(4 21 14)(5 22 15)(6 23 12)(7 20 13)(8 26 17)(9 27 18)(10 24 19)(11 25 16)
(1 24 17)(2 10 26)(3 19 8)(4 25 7)(5 9 23)(6 15 18)(11 13 14)(12 22 27)(16 20 21)
(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 24 6 26)(5 27 7 25)(8 14 10 12)(9 13 11 15)(16 22 18 20)(17 21 19 23)

G:=sub<Sym(27)| (1,16,27)(2,25,9)(3,11,18)(4,22,8)(5,17,14)(6,24,7)(10,13,12)(15,26,21)(19,20,23), (1,3,2)(4,21,14)(5,22,15)(6,23,12)(7,20,13)(8,26,17)(9,27,18)(10,24,19)(11,25,16), (1,24,17)(2,10,26)(3,19,8)(4,25,7)(5,9,23)(6,15,18)(11,13,14)(12,22,27)(16,20,21), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,24,6,26)(5,27,7,25)(8,14,10,12)(9,13,11,15)(16,22,18,20)(17,21,19,23)>;

G:=Group( (1,16,27)(2,25,9)(3,11,18)(4,22,8)(5,17,14)(6,24,7)(10,13,12)(15,26,21)(19,20,23), (1,3,2)(4,21,14)(5,22,15)(6,23,12)(7,20,13)(8,26,17)(9,27,18)(10,24,19)(11,25,16), (1,24,17)(2,10,26)(3,19,8)(4,25,7)(5,9,23)(6,15,18)(11,13,14)(12,22,27)(16,20,21), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,24,6,26)(5,27,7,25)(8,14,10,12)(9,13,11,15)(16,22,18,20)(17,21,19,23) );

G=PermutationGroup([(1,16,27),(2,25,9),(3,11,18),(4,22,8),(5,17,14),(6,24,7),(10,13,12),(15,26,21),(19,20,23)], [(1,3,2),(4,21,14),(5,22,15),(6,23,12),(7,20,13),(8,26,17),(9,27,18),(10,24,19),(11,25,16)], [(1,24,17),(2,10,26),(3,19,8),(4,25,7),(5,9,23),(6,15,18),(11,13,14),(12,22,27),(16,20,21)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,24,6,26),(5,27,7,25),(8,14,10,12),(9,13,11,15),(16,22,18,20),(17,21,19,23)])

G:=TransitiveGroup(27,83);

SU3(𝔽2) is a maximal subgroup of   He3⋊SD16
SU3(𝔽2) is a maximal quotient of   C2.SU3(𝔽2)

Matrix representation of SU3(𝔽2) in GL3(𝔽7) generated by

063
020
115
,
400
040
004
,
200
303
115
,
623
035
054
,
115
054
042
G:=sub<GL(3,GF(7))| [0,0,1,6,2,1,3,0,5],[4,0,0,0,4,0,0,0,4],[2,3,1,0,0,1,0,3,5],[6,0,0,2,3,5,3,5,4],[1,0,0,1,5,4,5,4,2] >;

SU3(𝔽2) in GAP, Magma, Sage, TeX

{\rm SU}_3({\mathbb F}_2)
% in TeX

G:=Group("SU(3,2)");
// GroupNames label

G:=SmallGroup(216,88);
// by ID

G=gap.SmallGroup(216,88);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,24,73,31,1347,297,543,6244,1330,916,382]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=c*b=b*c,e*a*e^-1=a*c,b*d=d*b,b*e=e*b,d*c*d^-1=a^-1,e*c*e^-1=a*b^-1*c^-1,e*d*e^-1=d^-1>;
// generators/relations

Export

Subgroup lattice of SU3(𝔽2) in TeX
Character table of SU3(𝔽2) in TeX

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