Finite Group Databasebeta

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#72235400

order4096
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C23×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72228242
commutator, Frattini, cosocleC22×H(3,𝔽4)C24
prime radical#72235019C22
LCSG ▷ C22×H(3,𝔽4) ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×H(3,𝔽4) ◁ G
LECSG ▷ C22×H(3,𝔽4) ▷ C24 ▷ C22 ▷ C1

6964 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 6964 conjugacy classes of subgroups — too many to draw. Only its 195 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=e,b^2=efh,c^2=egjkl,d^2=egjkl,e^2=l,g^2=kl,b^a=bef,c^a=chil,c^b=cg,d^a=dgj,d^b=dfhjl,d^c=degjkl,e^b=ei,e^c=eij,e^d=ejkl,f^a=fi,f^b=fijl,f^c=fijl,f^d=fj,f^e=fl,g^a=gjk,g^b=gjl,g^c=gijl,g^d=gjl,g^e=gk,g^f=gkl,h^a=hijl,h^b=hjl,h^c=hjkl,h^d=hi,h^e=hkl,h^g=hk,i^a=il,i^b=ik,i^c=ikl,i^d=il,j^a=jk,j^b=jkl,j^c=jl,j^d=jk

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72235419
Fitting subgroup of#72235419
maximal subgroup of#72235419
maximal normal subgroup of#72235419

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