Finite Group Databasebeta

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#72266433

order6144
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72266429 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC22#72209707
commutator, Fitting, cosocle, hypocenter#72266429C3
FrattiniC27C22⋊A4
socleC24#73292
LCSG ▷ #72266429
UCSC1 ◁ C22

27565 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 27565 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 96

Polycyclic presentation:

3,2^11|b^2=f,c^2=fgh,d^2=fgj,e^2=fjk,b^a=bc,c^a=bhk,c^b=cfg,d^a=de,d^b=dgj,d^c=dgi,e^a=dgh,e^b=eghijl,e^c=eghjk,e^d=efghjl,f^a=fhl,f^c=fk,f^d=fl,f^e=fl,g^a=ghkl,g^b=gk,g^c=gkl,g^d=gk,g^e=gkl,h^a=gl,h^b=hkl,h^c=hl,h^d=hkl,h^e=hl,i^a=gik,i^b=ikl,i^c=ikl,i^d=ik,j^a=ghjkl,j^b=jkl,j^c=jl,j^d=jkl,j^e=jl,k^a=kl,l^a=k

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72266494

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