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## G = CSU2(𝔽5)  order 240 = 24·3·5

### Conformal special unitary group on 𝔽52

Aliases: CSU2(𝔽5), C2.2S5, SL2(𝔽5).C2, SmallGroup(240,89)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — SL2(𝔽5) — CSU2(𝔽5)
 Derived series SL2(𝔽5) — CSU2(𝔽5)
 Lower central SL2(𝔽5) — CSU2(𝔽5)
 Upper central C1 — C2

10C3
6C5
10C4
15C4
10C6
6C10
5Q8
15Q8
15C8
10Dic3
10Dic3
10C12
6Dic5
15Q16
10Dic6

Character table of CSU2(𝔽5)

 class 1 2 3 4A 4B 5 6 8A 8B 10 12A 12B size 1 1 20 20 30 24 20 30 30 24 20 20 ρ1 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 linear of order 2 ρ3 4 4 1 -2 0 -1 1 0 0 -1 1 1 orthogonal lifted from S5 ρ4 4 4 1 2 0 -1 1 0 0 -1 -1 -1 orthogonal lifted from S5 ρ5 4 -4 -2 0 0 -1 2 0 0 1 0 0 symplectic faithful, Schur index 2 ρ6 4 -4 1 0 0 -1 -1 0 0 1 √3 -√3 symplectic faithful, Schur index 2 ρ7 4 -4 1 0 0 -1 -1 0 0 1 -√3 √3 symplectic faithful, Schur index 2 ρ8 5 5 -1 -1 1 0 -1 1 1 0 -1 -1 orthogonal lifted from S5 ρ9 5 5 -1 1 1 0 -1 -1 -1 0 1 1 orthogonal lifted from S5 ρ10 6 6 0 0 -2 1 0 0 0 1 0 0 orthogonal lifted from S5 ρ11 6 -6 0 0 0 1 0 -√2 √2 -1 0 0 symplectic faithful, Schur index 2 ρ12 6 -6 0 0 0 1 0 √2 -√2 -1 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of CSU2(𝔽5)
On 48 points
Generators in S48
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 22 5 18)(2 31 6 27)(3 8 7 4)(9 21 13 17)(10 32 14 28)(11 16 15 12)(19 35 23 39)(20 48 24 44)(25 41 29 45)(26 36 30 40)(33 38 37 34)(42 47 46 43)

G:=sub<Sym(48)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,22,5,18)(2,31,6,27)(3,8,7,4)(9,21,13,17)(10,32,14,28)(11,16,15,12)(19,35,23,39)(20,48,24,44)(25,41,29,45)(26,36,30,40)(33,38,37,34)(42,47,46,43)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,22,5,18)(2,31,6,27)(3,8,7,4)(9,21,13,17)(10,32,14,28)(11,16,15,12)(19,35,23,39)(20,48,24,44)(25,41,29,45)(26,36,30,40)(33,38,37,34)(42,47,46,43) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,22,5,18),(2,31,6,27),(3,8,7,4),(9,21,13,17),(10,32,14,28),(11,16,15,12),(19,35,23,39),(20,48,24,44),(25,41,29,45),(26,36,30,40),(33,38,37,34),(42,47,46,43)]])

CSU2(𝔽5) is a maximal subgroup of   C4.6S5  C4.S5  C22.S5
CSU2(𝔽5) is a maximal quotient of   C22.2S5

Matrix representation of CSU2(𝔽5) in GL4(𝔽3) generated by

 1 2 0 1 0 0 1 2 2 0 2 2 2 2 1 0
,
 2 0 2 1 1 2 0 2 0 2 0 1 1 2 2 2
G:=sub<GL(4,GF(3))| [1,0,2,2,2,0,0,2,0,1,2,1,1,2,2,0],[2,1,0,1,0,2,2,2,2,0,0,2,1,2,1,2] >;

CSU2(𝔽5) in GAP, Magma, Sage, TeX

{\rm CSU}_2({\mathbb F}_5)
% in TeX

G:=Group("CSU(2,5)");
// GroupNames label

G:=SmallGroup(240,89);
// by ID

G=gap.SmallGroup(240,89);
# by ID

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