Finite Group Databasebeta

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#72340916

order6144
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72241389 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#72241389C3
FrattiniC25C22⋊(C22⋊A4)
socleC22#72227294
LCSG ▷ #72241389
UCSC1

9971 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 9971 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 48

Polycyclic presentation:

3,2^11|b^2=i,c^2=il,d^2=hikl,e^2=hil,f^2=k,g^2=kl,b^a=ck,c^a=bcik,c^b=cik,d^a=ekl,d^b=dj,d^c=dh,e^a=dehi,e^b=ehj,e^c=ejkl,e^d=ehi,f^a=gk,f^b=fk,f^c=fikl,f^d=fikl,f^e=fj,g^a=fg,g^b=gikl,g^c=gkl,g^d=gijl,g^e=gi,h^a=hl,h^b=hl,h^c=hk,h^d=hk,h^e=hkl,h^f=hk,h^g=hkl,i^a=il,i^d=ik,i^e=ikl,j^a=jkl,j^b=jk,j^c=jkl,j^d=jl,j^e=jk,k^a=kl,l^a=k

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72340935
maximal subgroup of#72340935
maximal normal subgroup of#72340935

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