Finite Group Databasebeta

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#72266425

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC23#93139
commutatorC22×#384C42
Frattini, prime radical, cosocle#72266422C22
LCSG ▷ C22×#384 ▷ C23×C42 ▷ C25 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C42 ◁ #72266422 ◁ G
LECSG ▷ #72266422 ▷ C23×C42 ▷ C25 ▷ C23 ▷ C1

6686 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 6686 conjugacy classes of subgroups — too many to draw. Only its 61 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,d^2=jkl,e^2=jkl,f^2=k,g^2=kl,b^a=be,c^b=cf,d^a=dg,d^c=dhk,e^a=efjl,e^b=egj,e^c=eik,e^d=ehi,f^a=fi,f^b=fh,f^c=fk,f^d=fjl,f^e=fk,g^a=ghl,g^b=ghikl,g^c=gj,g^d=gkl,g^e=gkl,h^a=hj,h^b=hjl,i^a=ik,i^b=ijk

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72266428
Fitting subgroup of#72266428
maximal subgroup of#72266428
maximal normal subgroup of#72266428

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