PΓL(2,𝔽₂₅) #72224463

order	31200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

projective general semilinear

PΓL(2,𝔽₂₅)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, cosocle, hypocenter	PSL(2,𝔽₂₅)	C₂²
LCS	G ▷ PSL(2,𝔽₂₅)
UCS	C₁

115 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 115 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 26

A minimal faithful permutation representation — 2 generators on 26 points:

(1,6,20,21,22,10)(2,13,9)(3,19,5,23,17,14)(4,15,8,16,25,12)(7,24,11)(18,26);(1,22,21,12,2,4,14,17,5,18,25,19,23,11,16,3,9,24,13,20,8,7,10,6,15,26)

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

automorphism group of

#72224461

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,6,20,21,22,10)(2,13,9)(3,19,5,23,17,14)(4,15,8,16,25,12)(7,24,11)(18,26), (1,22,21,12,2,4,14,17,5,18,25,19,23,11,16,3,9,24,13,20,8,7,10,6,15,26));