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#72267784

order4096
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC23#125654
commutator, Frattini, cosocleC2×#7555C23
prime radicalC2×#158995C22
LCSG ▷ C2×#7555 ▷ C26 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C26 ◁ C27 ◁ C2×#7555 ◁ G
LECSG ▷ C2×#7555 ▷ C26 ▷ C23 ▷ C22 ▷ C1

33356 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 33356 conjugacy classes of subgroups — too many to draw. Only its 229 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) = 64

Polycyclic presentation:

2^12|a^2=j,b^2=f,c^2=fhl,d^2=gk,e^2=hj,b^a=bd,c^a=cdehj,c^b=cfgjl,d^a=dg,d^b=dgjk,d^c=di,e^a=eh,e^b=egijk,e^c=eghijkl,e^d=ejl,f^a=fj,f^c=fk,f^d=fk,f^e=fl,g^c=gk,h^b=hkl,h^c=hk,i^a=ij,i^b=ikl,i^c=ik,i^d=ik,i^e=il,j^b=jk,j^c=jkl

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72267786
maximal subgroup of#72267786
maximal normal subgroup of#72267786

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