Finite Group Databasebeta

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#72209976

order2048
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C29

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2×C42H(3,𝔽4)
commutator, Frattini, cosocleC23×C42C24
socleC23#15947
prime radicalC2×#62694C22
LCSG ▷ C23×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C42 ◁ C23×C42 ◁ G
LECSG ▷ C23×C42 ▷ C22 ▷ C1

14062 conjugacy classes of subgroups; only the 716 normal subgroups are shown.

normal subgroups only

This group has 14062 conjugacy classes of subgroups — too many to draw. Only its 716 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:

2^11|c^2=e,d^2=eghik,e^2=k,f^2=k,g^2=jk,h^2=jk,i^2=j,c^a=cg,c^b=ci,d^a=df,d^b=dh,d^c=defghjk,e^a=ej,e^b=ejk,f^a=fk,f^b=fj,f^c=fk,f^d=fjk,g^a=gjk,g^b=gk,g^c=gk,g^d=gjk,h^a=hj,h^b=hjk,i^a=ik,i^b=ij,i^c=ik,i^d=ijk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209979
maximal normal subgroup of#72209979

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