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G = SL2(𝔽7)  order 336 = 24·3·7

Special linear group on 𝔽72

non-abelian, perfect, quasisimple, not soluble

Aliases: SL2(𝔽7), SU2(𝔽7), Spin3(𝔽7), C2.GL3(𝔽2), SmallGroup(336,114)

Series: ChiefDerived Lower central Upper central

C1C2 — SL2(𝔽7)
SL2(𝔽7)
SL2(𝔽7)
C1C2

28C3
8C7
21C4
28C6
8C14
8C7⋊C3
7Q8
7Q8
21C8
28Dic3
8C2×C7⋊C3
21Q16
7SL2(𝔽3)
7SL2(𝔽3)
7CSU2(𝔽3)
7CSU2(𝔽3)

Character table of SL2(𝔽7)

 class 123467A7B8A8B14A14B
 size 11564256242442422424
ρ111111111111    trivial
ρ2330-10-1--7/2-1+-7/211-1--7/2-1+-7/2    complex lifted from GL3(𝔽2)
ρ3330-10-1+-7/2-1--7/211-1+-7/2-1--7/2    complex lifted from GL3(𝔽2)
ρ44-410-11+-7/21--7/200-1--7/2-1+-7/2    complex faithful
ρ54-410-11--7/21+-7/200-1+-7/2-1--7/2    complex faithful
ρ666020-1-100-1-1    orthogonal lifted from GL3(𝔽2)
ρ76-6000-1-12-211    symplectic faithful, Schur index 2
ρ86-6000-1-1-2211    symplectic faithful, Schur index 2
ρ9771-1100-1-100    orthogonal lifted from GL3(𝔽2)
ρ1088-10-1110011    orthogonal lifted from GL3(𝔽2)
ρ118-8-1011100-1-1    symplectic faithful, Schur index 2

Permutation representations of SL2(𝔽7)
On 16 points - transitive group 16T715
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13 8)(4 5 9)(10 12 15)(11 14 16)

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

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

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

G:=TransitiveGroup(16,715);

Matrix representation of SL2(𝔽7) in GL2(𝔽7) generated by

54
26
,
04
56
G:=sub<GL(2,GF(7))| [5,2,4,6],[0,5,4,6] >;

SL2(𝔽7) in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_7)
% in TeX

G:=Group("SL(2,7)");
// GroupNames label

G:=SmallGroup(336,114);
// by ID

G=gap.SmallGroup(336,114);
# by ID

Export

Subgroup lattice of SL2(𝔽7) in TeX
Character table of SL2(𝔽7) in TeX

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