Finite Group Databasebeta

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#72222976

order15552
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23×C3

Aliases

No aliases for this group.

Splittings

FittingC24×C33 ⋊ D32
derived#72220499 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72220499C22
FrattiniC32#13181790
FittingC24×C33D32
socleC24×C32#1113
LCSG ▷ #72220499
UCSC1

3495 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 3495 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

Polycyclic presentation:

2^2,3^5,2^4|c^a=c^2,d^a=d^2,d^b=d^2,d^c=df,e^a=d^2e,e^b=e^2,e^c=efg^2,f^b=f^2,g^a=fg^2,g^b=g^2,h^d=i,h^e=hi,i^a=hi,i^b=hi,i^d=hi,i^e=h,j^a=hj,j^b=hj,j^d=hj,j^e=k,k^b=ijk,k^d=hik,k^e=jk

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226353
maximal normal subgroup of#72226353

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