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#72277708

order4374
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14551735 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#6866C2×C32
FrattiniC34C32×D3
Fitting#14551735C2
socleC3#13091189
LCSG ▷ #6866
UCSC1

373 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 30 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:

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

Polycyclic presentation:

2,3^7|d^3=g,c^b=cf^2h,d^a=d^2g^2,d^b=de,d^c=dg^2h,e^a=e^2fh^2,e^b=eh^2,e^c=eh,e^d=ef,f^a=fh,f^d=fh,g^a=g^2,g^b=gh,h^a=h^2

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72277725

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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6^2*F.8);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7^2*F.8);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6*F.8^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.8^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);