PΣL(2,𝔽₂₇) #72224464

order	29484
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

projective special semilinear

PΣL(2,𝔽₂₇)

Splittings

socle, derived

PSL(2,𝔽₂₇) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, cosocle, hypocenter	PSL(2,𝔽₂₇)	C₃
LCS	G ▷ PSL(2,𝔽₂₇)
UCS	C₁

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 28

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,5,12,11,19,4,8,18,7,10,23,24)(3,9,16,25,20,13,21,27,6,14,15,26,28);(2,6,14)(3,10,15)(4,11,24)(7,13,26)(9,21,18)(16,20,19)(17,22,27)(23,25,28)

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	PΓL(2,𝔽₂₇)
maximal normal subgroup of	PΓL(2,𝔽₂₇)
maximal quotient of	ΣL(2,𝔽₂₇)

GAP

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

gap> G := Group((1,2,5,12,11,19,4,8,18,7,10,23,24)(3,9,16,25,20,13,21,27,6,14,15,26,28), (2,6,14)(3,10,15)(4,11,24)(7,13,26)(9,21,18)(16,20,19)(17,22,27)(23,25,28));

PΣL(2,𝔽27) #72224464