Finite Group Databasebeta

← Prev (#72259772) Next (#72259774) →

#72259773

order5832
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C34

Aliases

No aliases for this group.

Splittings

FittingC22×3−1+4 ⋊ D3
derivedC3×C9×A4 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C9×A4C2×C32
FrattiniC3C32×C3⋊(C3⋊S4)
FittingC22×3−1+4D3
socleC22×C3C32×C3⋊(C3⋊D3)
LCSG ▷ C3×C9×A4
UCSC1

3631 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 3631 conjugacy classes of subgroups — too many to draw. Only its 215 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) = 108

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

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

Polycyclic presentation:

2,3^6,2^2|f^3=g,b^a=bg^2,d^a=d^2,d^b=dg^2,e^a=e^2,f^a=f^2g^2,f^b=fg^2,f^c=fg,g^a=g^2,h^a=hi,h^e=i,i^e=hi

Conjugacy classes & character table

193 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, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);