Finite Group Databasebeta

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#72266360

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC23#93139
commutatorC22×#384C42
Frattini, cosocle#72266358C22
prime radicalC25#1304
LCSG ▷ C22×#384 ▷ C22×C42 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C42 ◁ #72266358 ◁ G
LECSG ▷ #72266358 ▷ C23×C42 ▷ C24 ▷ C22 ▷ C1

7042 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 7042 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 256

Polycyclic presentation:

2^12|a^2=c,b^2=d,c^2=h,d^2=h,e^2=h,f^2=kl,g^2=k,b^a=be,c^b=cf,d^a=dg,d^c=dikl,e^a=efhkl,e^b=eghk,e^c=ejkl,e^d=eij,f^a=fj,f^b=fi,f^c=fkl,f^d=fkl,f^e=fkl,g^a=gil,g^b=gijk,g^c=gk,g^d=gk,g^e=gk,i^a=ik,i^b=ikl,j^a=jkl,j^b=jl

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72266363
Fitting subgroup of#72266363
maximal subgroup of#72266363
maximal normal subgroup of#72266363

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