Finite Group Databasebeta

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#72241390

order6144
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72241389 ⋊ C3
derived#8934620 ⋊ C22×C3

Subgroups

subgroupG/H
center, socleC22#72216883
commutator, cosocle, hypocenter#8934620C22×C3
FrattiniC25C22×C22⋊A4
Fitting#72241389C3
hypercenterC23#11710729
LCSG ▷ #8934620
UCSC1 ◁ C22 ◁ C23

10081 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 10081 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 128

Polycyclic presentation:

2^2,3,2^9|d^2=k,e^2=k,f^2=l,g^2=l,d^a=dhi,d^b=dhi,d^c=e,e^a=eh,e^b=eh,e^c=dek,e^d=ek,f^a=fhikl,f^c=g,f^e=fj,g^a=ghkl,g^c=fgl,g^d=gj,g^f=gl,h^c=ik,h^d=hk,h^f=hl,i^c=hik,i^d=ik,i^e=ik,i^f=il,i^g=il,j^a=jkl,j^b=jl

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252372, #72252390, #72252407
maximal normal subgroup of#72252372, #72252390, #72252407

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