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PGL(2,𝔽23) #72210372

order12144
automorphismAut(G) = PGL(2,𝔽23) (order 12144) · Inn(G) = G · Out(G) = C1
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2

Aliases

projective general linearPGL(2,𝔽23)

Splittings

socle, derivedPSL(2,𝔽23) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, socle, cosocle, hypocenterPSL(2,𝔽23)C2
LCSG ▷ PSL(2,𝔽23)
UCSC1

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

automorphism group ofPSL(2,𝔽23), Sp(2,𝔽23), PGL(2,𝔽23)

GAP

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

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