ΣL(2,𝔽₂₇) #72224469

order	58968
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

special semilinear

ΣL(2,𝔽₂₇)

Splittings

derived

SL(2,𝔽₂₇) ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, Fitting, socle, hypercenter	C₂	PΣL(2,𝔽₂₇)
commutator, cosocle, hypocenter	SL(2,𝔽₂₇)	C₃
LCS	G ▷ SL(2,𝔽₂₇)
UCS	C₁ ◁ C₂

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 56 points:

(1,41,33,11,49,26,22,4,53,45,56,21)(2,16,8,12,44,34,14,24,47,20,40,32)(3,10,19,6,37,55,18,39,36,5,17,50)(7,29,30,46,54,27,43,25,31,51,35,23)(9,15,48,52)(13,38,28,42);(1,42,44,22,38,40)(2,52,54,14,15,35)(3,17,34,18,37,32)(4,9,33,41,48,53)(5,20,23,6,12,27)(7,50,39,43,55,10)(8,51,49,47,46,56)(11,19,24,45,36,16)(13,25,31,28,29,30)(21,26)

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

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

gap> G := Group((1,41,33,11,49,26,22,4,53,45,56,21)(2,16,8,12,44,34,14,24,47,20,40,32)(3,10,19,6,37,55,18,39,36,5,17,50)(7,29,30,46,54,27,43,25,31,51,35,23)(9,15,48,52)(13,38,28,42), (1,42,44,22,38,40)(2,52,54,14,15,35)(3,17,34,18,37,32)(4,9,33,41,48,53)(5,20,23,6,12,27)(7,50,39,43,55,10)(8,51,49,47,46,56)(11,19,24,45,36,16)(13,25,31,28,29,30)(21,26));

ΣL(2,𝔽27) #72224469