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## G = CSU2(𝔽3)  order 48 = 24·3

### Conformal special unitary group on 𝔽32

Aliases: CSU2(𝔽3), Q8.S3, C2.2S4, SL2(𝔽3).C2, SmallGroup(48,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — CSU2(𝔽3)
 Lower central SL2(𝔽3) — CSU2(𝔽3)
 Upper central C1 — C2

Generators and relations for CSU2(𝔽3)
G = < a,b,c,d | a4=c3=1, b2=d2=a2, bab-1=dbd-1=a-1, cac-1=ab, dad-1=a2b, cbc-1=a, dcd-1=c-1 >

Character table of CSU2(𝔽3)

 class 1 2 3 4A 4B 6 8A 8B size 1 1 8 6 12 8 6 6 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 -1 -1 linear of order 2 ρ3 2 2 -1 2 0 -1 0 0 orthogonal lifted from S3 ρ4 2 -2 -1 0 0 1 √2 -√2 symplectic faithful, Schur index 2 ρ5 2 -2 -1 0 0 1 -√2 √2 symplectic faithful, Schur index 2 ρ6 3 3 0 -1 -1 0 1 1 orthogonal lifted from S4 ρ7 3 3 0 -1 1 0 -1 -1 orthogonal lifted from S4 ρ8 4 -4 1 0 0 -1 0 0 symplectic faithful, Schur index 2

Permutation representations of CSU2(𝔽3)
On 16 points - transitive group 16T65
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5 3 7)(2 8 4 6)(9 14 11 16)(10 13 12 15)
(2 5 8)(4 7 6)(9 15 14)(11 13 16)
(1 10 3 12)(2 15 4 13)(5 9 7 11)(6 16 8 14)

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

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

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

G:=TransitiveGroup(16,65);

Matrix representation of CSU2(𝔽3) in GL2(𝔽7) generated by

 2 1 2 5
,
 1 2 6 6
,
 4 0 2 2
,
 2 5 6 5
G:=sub<GL(2,GF(7))| [2,2,1,5],[1,6,2,6],[4,2,0,2],[2,6,5,5] >;

CSU2(𝔽3) in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(48,28);
// by ID

G=gap.SmallGroup(48,28);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-2,120,41,182,277,72,123,188,133,58]);
// Polycyclic

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

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