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AΓL(1,𝔽81) #72224506

order25920
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C1

Aliases

affine general semilinearAΓL(1,𝔽81)

Splittings

Fitting, socle, SylowC34 ⋊ ΓL(1,𝔽81)
HallC34⋊C5 ⋊ #365

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator#72210626C2×C4
Fitting, socleC34ΓL(1,𝔽81)
prime radical#11717694⋊C2SD16
cosocle#72224478C22
hypocenterC34⋊C5#365
LCSG ▷ #72210626 ▷ #66472⋊C5 ▷ (C3⋊(C3⋊(C3⋊D3)))⋊C5 ▷ C34⋊C5
UCSC1

127 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 127 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 2 generators on 81 points:

(1,53,5,79,43,60,81,17,42,41,77,54)(2,12,48,11,68,13,61,70,27,36,49,35)(3,24,52,50,39,58,73,34,4,30,72,25)(6,44,15,20,56,46,18,71,78,51,55,37)(7,21,26,22,19,9)(8,23,14,64,63,16,75,32,62,45,33,47)(10,38,67,80,66,59,74,28,40,57,65,76)(29,69,31);(2,32,67,28,66,23,39,73,58,30,20,80,69,4,18,77,12,42,57,64,10,71,79,75,8,13,53,52,16,27,50,36,59,25,19,34,35,72,37,5,11,55,76,74,17,62,7,51,14,33,9,65,81,26,47,29,56,43,60,15,54,21,40,24,45,6,38,44,63,70,49,78,41,48,3,61,68,46,22,31)

Polycyclic presentation:

2^6,5,3^4|a^2=d,b^2=c,d^2=e,e^2=f,b^a=bdef,c^a=cf,d^b=de,e^b=ef,g^b=g^3,g^c=g^4,h^a=hi^2j^2,h^b=hij^2,h^c=h^2jk^2,h^d=h^2j^2k^2,h^e=h^2jk,h^f=h^2,h^g=hk,i^a=h^2i^2j^2,i^d=i^2jk^2,i^e=i^2j^2k,i^f=i^2,i^g=hi,j^a=jk,j^b=h^2i^2jk,j^c=ijk,j^d=hi^2,j^e=h^2ij,j^f=j^2,j^g=hij^2k^2,k^a=hi^2j^2k^2,k^b=h^2ij^2,k^c=ik^2,k^d=hi,k^e=h^2i^2k,k^f=k^2,k^g=hij^2k

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

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", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 5, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8*F.9^2*F.10^2);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10^2);;
gap> SetConjugate(coll, 8, 3, F.8^2*F.10*F.11^2);;
gap> SetConjugate(coll, 8, 4, F.8^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 8, 5, F.8^2*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.8^2);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9^2*F.10^2);;
gap> SetConjugate(coll, 9, 4, F.9^2*F.10*F.11^2);;
gap> SetConjugate(coll, 9, 5, F.9^2*F.10^2*F.11);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 9, 7, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.8^2*F.9^2*F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.8*F.9^2);;
gap> SetConjugate(coll, 10, 5, F.8^2*F.9*F.10);;
gap> SetConjugate(coll, 10, 6, F.10^2);;
gap> SetConjugate(coll, 10, 7, F.8*F.9*F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 1, F.8*F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 2, F.8^2*F.9*F.10^2);;
gap> SetConjugate(coll, 11, 3, F.9*F.11^2);;
gap> SetConjugate(coll, 11, 4, F.8*F.9);;
gap> SetConjugate(coll, 11, 5, F.8^2*F.9^2*F.11);;
gap> SetConjugate(coll, 11, 6, F.11^2);;
gap> SetConjugate(coll, 11, 7, F.8*F.9*F.10^2*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);