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## G = SU3(𝔽2)  order 216 = 23·33

### Special unitary group on 𝔽23

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

 Derived series C1 — C3 — He3 — He3⋊C2 — SU3(𝔽2)
 Chief series C1 — C3 — He3 — He3⋊C2 — He3⋊C4 — SU3(𝔽2)
 Lower central He3 — He3⋊C2 — SU3(𝔽2)
 Upper central C1 — C3

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 >

Character table of SU3(𝔽2)

 class 1 2 3A 3B 3C 4A 4B 4C 6A 6B 12A 12B 12C 12D 12E 12F size 1 9 1 1 24 18 18 18 9 9 18 18 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ5 2 -2 2 2 2 0 0 0 -2 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ6 3 -1 -3-3√-3/2 -3+3√-3/2 0 -1 1 -1 ζ6 ζ65 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 complex faithful ρ7 3 -1 -3+3√-3/2 -3-3√-3/2 0 1 1 1 ζ65 ζ6 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 complex faithful ρ8 3 -1 -3-3√-3/2 -3+3√-3/2 0 -1 -1 1 ζ6 ζ65 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 complex faithful ρ9 3 -1 -3+3√-3/2 -3-3√-3/2 0 -1 1 -1 ζ65 ζ6 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 complex faithful ρ10 3 -1 -3-3√-3/2 -3+3√-3/2 0 1 1 1 ζ6 ζ65 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 complex faithful ρ11 3 -1 -3+3√-3/2 -3-3√-3/2 0 -1 -1 1 ζ65 ζ6 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 complex faithful ρ12 3 -1 -3+3√-3/2 -3-3√-3/2 0 1 -1 -1 ζ65 ζ6 ζ3 ζ6 ζ65 ζ65 ζ32 ζ6 complex faithful ρ13 3 -1 -3-3√-3/2 -3+3√-3/2 0 1 -1 -1 ζ6 ζ65 ζ32 ζ65 ζ6 ζ6 ζ3 ζ65 complex faithful ρ14 6 2 -3-3√-3 -3+3√-3 0 0 0 0 -1-√-3 -1+√-3 0 0 0 0 0 0 complex faithful ρ15 6 2 -3+3√-3 -3-3√-3 0 0 0 0 -1+√-3 -1-√-3 0 0 0 0 0 0 complex faithful ρ16 8 0 8 8 -1 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from PSU3(𝔽2)

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

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

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

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

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

 0 6 3 0 2 0 1 1 5
,
 4 0 0 0 4 0 0 0 4
,
 2 0 0 3 0 3 1 1 5
,
 6 2 3 0 3 5 0 5 4
,
 1 1 5 0 5 4 0 4 2
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

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