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PΓL(2,𝔽27) #72224466

order58968
automorphismAut(G) = PΓL(2,𝔽27) (order 58968) · Inn(G) = G · Out(G) = C1
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

projective general semilinearPΓL(2,𝔽27)

Splittings

socle, derivedPSL(2,𝔽27) ⋊ C2×C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, socle, cosocle, hypocenterPSL(2,𝔽27)C2×C3
LCSG ▷ PSL(2,𝔽27)
UCSC1

67 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 28

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

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

Conjugacy classes & character table

23 conjugacy classes — view the full character table.

Relationships

automorphism group ofPΓL(2,𝔽27)

GAP

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

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