Finite Group Databasebeta

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#72235024

order4096
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C29

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72209716
commutator, Frattini, cosocleC22×H(3,𝔽4)C24
LCSG ▷ C22×H(3,𝔽4) ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×H(3,𝔽4) ◁ G
LECSG ▷ C22×H(3,𝔽4) ▷ C24 ▷ C22 ▷ C1

120971 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 120971 conjugacy classes of subgroups — too many to draw. Only its 309 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=k,b^2=l,c^2=jkl,d^2=ij,e^2=l,f^2=l,g^2=kl,h^2=kl,c^a=cf,c^b=ch,d^a=de,d^b=dg,e^a=el,e^b=ek,e^c=ei,e^d=ek,f^a=fl,f^b=fk,f^c=fkl,f^d=fil,f^e=fl,g^a=gk,g^b=gkl,g^c=gj,g^d=gk,g^f=gk,h^a=hk,h^b=hkl,h^d=hjkl,h^e=hk,h^g=hkl,i^a=il,i^b=ik,j^a=jk,j^b=jkl

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72235031
Fitting subgroup of#72235031, #72268534
maximal subgroup of#72235031, #72268534
maximal normal subgroup of#72235031, #72268534

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