Finite Group Databasebeta

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#72259999

order5832
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

FittingC22×#6891 ⋊ D3
derived#72228407 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72228407C2
FrattiniC33C3⋊(C3⋊S4)
FittingC22×#6891D3
socleC22×C3#92338
LCSG ▷ #72228407
UCSC1

903 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 903 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 31
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 4 generators on 31 points:

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

Polycyclic presentation:

2,3^6,2^2|b^3=g,d^3=g,b^a=b^2g^2,c^a=c^2,c^b=ce,d^a=d^2g^2,d^c=de^2f,e^d=eg,f^a=fg^2,f^b=fg^2,f^d=fg^2,g^a=g^2,h^a=hi,h^d=i,i^d=hi

Conjugacy classes & character table

81 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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 3, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.7^2);;
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*F.7^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.5^2*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);