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#72277689

order4374
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C3

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#14547000 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#14547000C2
FrattiniC34C3⋊(C3⋊D3)
socleC32#92338
LCSG ▷ #14547000
UCSC1

1112 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 36
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 4 generators on 36 points:

(2,5)(3,4)(6,9)(7,8)(11,14)(12,18)(13,21)(15,24)(16,26)(17,23)(19,30)(20,31)(22,28)(25,33)(27,35)(29,34)(32,36);(10,12,18)(11,16,28)(13,20,30)(14,22,26)(15,25,34)(17,27,36)(19,31,21)(23,32,35)(24,29,33);(2,6,7)(5,8,9)(16,27,29)(22,32,25)(26,34,35)(28,33,36);(1,2,5)(3,6,8)(4,7,9)(10,17,14,13,24,15,21,11,23)(12,27,22,20,29,25,19,16,32)(18,36,26,30,33,34,31,28,35)

Polycyclic presentation:

2,3^7|b^3=h,b^a=b^2h^2,c^a=c^2,c^b=ce,d^a=d^2,d^c=df,e^a=eg^2,e^b=eg,e^d=eh,f^b=fh^2,g^a=g^2,h^a=h^2

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72277725
maximal normal 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, 2, F.8);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.8^2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);