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#72260073

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

Aliases

No aliases for this group.

Splittings

FittingC22×#6910 ⋊ D3
derived#72228413 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72228413C2
FrattiniC33C3⋊(C3⋊S4)
FittingC22×#6910D3
socleC22×C3#92338
LCSG ▷ #72228413
UCSC1

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

normal subgroups only

This group has 834 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)(6,11)(7,17)(8,19)(9,10)(12,16)(13,26)(14,22)(15,24)(18,30)(20,21)(23,28)(25,29)(27,31);(2,3,4)(5,21,19,9,17,7,10,8,20)(6,27,28,15,29,18,16,14,13)(11,26,22,12,30,25,24,23,31);(5,20,17,9,19,8,10,7,21)(6,28,14,15,18,27,16,13,29)(11,25,26,12,31,30,24,22,23);(5,15,11,9,16,12,10,6,24)(7,13,25,20,28,31,19,18,22)(8,27,23,21,29,26,17,14,30)

Polycyclic presentation:

2,3^6,2^2|b^3=g^2,c^3=g^2,d^3=g^2,b^a=b^2g,c^a=c^2g,c^b=ce,d^a=d^2g,d^c=de^2f,e^a=eg^2,e^c=eg,e^d=eg^2,f^a=fg^2,f^b=fg,f^c=fg^2,f^d=fg,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, 2, F.7^2);;
gap> SetPower(coll, 3, F.7^2);;
gap> SetPower(coll, 4, F.7^2);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5^2*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 4, 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, 4, F.9);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);