Finite Group Databasebeta

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#72268229

order4096
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C211

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72209716
commutator, Frattini, cosocleC22×#543C24
LCSG ▷ C22×#543 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#543 ◁ G
LECSG ▷ C22×#543 ▷ C24 ▷ C22 ▷ C1

Only the 323 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 323 normal subgroups are shown; non-normal subgroups are omitted.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^12|b^2=l,c^2=ijl,d^2=ijl,e^2=l,f^2=l,c^a=cg,c^b=ce,d^a=dgh,d^b=df,d^c=dijl,e^a=ek,e^b=el,e^c=ek,e^d=ej,f^a=fk,f^b=fl,f^c=fjk,f^d=fk,f^e=fl,g^b=gk,g^c=gk,g^d=gi,g^f=gk,h^c=hi,h^d=hik,h^e=hk,h^f=hk,i^b=ik,j^a=jk,j^b=jl

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72271332
maximal subgroup of#72271332
maximal normal subgroup of#72271332

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.12);;
gap> SetPower(coll, 3, F.9*F.10*F.12);;
gap> SetPower(coll, 4, F.9*F.10*F.12);;
gap> SetPower(coll, 5, F.12);;
gap> SetPower(coll, 6, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10*F.12);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);