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AΓL(1,𝔽27) #72224465

order2106
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

affine general semilinearAΓL(1,𝔽27)

Splittings

Fitting, socleC33 ⋊ C2×C13⋊C3
derivedC33⋊C13 ⋊ C2×C3
Hall(C33⋊C13)⋊C3 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC33⋊C13C2×C3
Fitting, socleC33C2×C13⋊C3
LCSG ▷ C33⋊C13
UCSC1

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 27

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

2,3,13,3^3|c^b=c^3,d^a=d^2,d^b=def,d^c=de,e^a=e^2,e^b=f^2,e^c=d^2ef,f^a=f^2,f^b=de^2f^2,f^c=d^2e

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofPΓL(2,𝔽27)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 3, 13, 3, 3, 3]);;
gap> SetConjugate(coll, 3, 2, F.3^3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.6^2);;
gap> SetConjugate(coll, 5, 3, F.4^2*F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.4*F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 3, F.4^2*F.5);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);