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#72227641

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72227638 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC22×C4#5464
commutator, Fitting, cosocle, hypocenter#72227638C3
FrattiniC24×C4C22⋊A4
socleC23#2679⋊C3
LCSG ▷ #72227638
UCSC1 ◁ C22×C4

1667 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 1667 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 32
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

3,2^10|b^2=f,c^2=f,d^2=fj,e^2=fj,g^2=k,i^2=k,b^a=bc,c^a=b,c^b=cf,d^a=e,d^b=dh,d^c=dhik,e^a=defj,e^b=eij,e^c=efgk,e^d=efj,g^a=fhij,g^c=gk,g^e=gk,h^a=fgijk,h^c=hk,h^e=hk,i^a=fgk,i^b=ik,i^d=ik,i^e=ik

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72227666
maximal subgroup of#72227666
maximal normal subgroup of#72227666

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", "j", "k");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.6*F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.2);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.11);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.6*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.6*F.7*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);