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PGL(2,𝔽19) #72210366

order6840
automorphismAut(G) = PGL(2,𝔽19) (order 6840) · Inn(G) = G · Out(G) = C1
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2

Aliases

projective general linearPGL(2,𝔽19)

Splittings

socle, derivedPSL(2,𝔽19) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, socle, cosocle, hypocenterPSL(2,𝔽19)C2
LCSG ▷ PSL(2,𝔽19)
UCSC1

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(1,20)(3,11)(4,14)(5,6)(7,17)(8,12)(9,13)(10,18)(15,16);(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19);(2,11,6,13,7,4,12,16,18,19,10,15,8,14,17,9,5,3)

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

automorphism group ofPSL(2,𝔽19), Sp(2,𝔽19), PGL(2,𝔽19)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((1,20)(3,11)(4,14)(5,6)(7,17)(8,12)(9,13)(10,18)(15,16), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,11,6,13,7,4,12,16,18,19,10,15,8,14,17,9,5,3));