Finite Group Databasebeta

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#72340920

order6144
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72241349 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#72241349C3
FrattiniC25C22⋊(C22⋊A4)
socleC22#72227294
LCSG ▷ #72241349
UCSC1

10541 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 10541 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=i,d^2=ijkl,e^2=ijkl,b^a=c,c^a=bci,c^b=ci,d^a=e,d^b=dh,d^c=dhj,e^a=deijkl,e^b=ejkl,e^c=ehk,e^d=eijkl,f^a=g,f^c=fi,f^d=fhj,f^e=fhijk,g^a=fg,g^b=gi,g^d=gik,g^e=ghjkl,h^a=hk,h^b=hk,h^c=hl,h^d=hl,h^e=hkl,h^f=hkl,h^g=hk,i^d=ik,i^e=il,j^a=jl,j^b=jl,j^c=jkl,j^d=jk,j^e=jl,j^f=jkl,j^g=jk,k^a=l,l^a=kl

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72340940, #72340957, #72340959, …
maximal subgroup of#72340940, #72340957, #72340959, …
maximal normal subgroup of#72340940, #72340957, #72340959, …

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