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#72259892

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

Aliases

No aliases for this group.

Splittings

FittingC22×#6909 ⋊ D3
derived#72228403 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72228403C2
FrattiniC33C3⋊(C3⋊S4)
FittingC22×#6909D3
socleC22×C3#92338
LCSG ▷ #72228403
UCSC1

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

normal subgroups only

This group has 842 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) = 108

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

73 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, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, 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);