Finite Group Databasebeta

← Prev (#72258121) Next (#72258123) →

#72258122

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72258096 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#72258096C3
FrattiniC24×C42C22⋊A4
socleC22#72238329
LCSG ▷ #72258096
UCSC1

57452 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 57452 conjugacy classes of subgroups — too many to draw. Only its 47 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:

3,2^12|b^2=f,c^2=g,d^2=fgiklm,e^2=fhm,f^2=l,g^2=m,h^2=l,j^2=m,k^2=lm,b^a=cfil,c^a=bci,c^b=ci,d^a=efgiklm,d^b=dijm,d^c=dijkm,e^a=de,e^b=eh,e^c=ehjl,f^a=gm,f^c=fl,g^a=fgil,g^b=gm,h^a=hikl,h^b=hl,h^c=hm,i^a=im,i^b=il,i^c=im,j^a=hijl,j^b=jlm,j^c=jl,k^a=h,k^b=kl,k^c=km,l^a=m,m^a=lm

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258131

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", "m");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.6*F.7*F.9*F.11*F.12*F.13);;
gap> SetPower(coll, 5, F.6*F.8*F.13);;
gap> SetPower(coll, 6, F.12);;
gap> SetPower(coll, 7, F.13);;
gap> SetPower(coll, 8, F.12);;
gap> SetPower(coll, 10, F.13);;
gap> SetPower(coll, 11, F.12*F.13);;
gap> SetConjugate(coll, 2, 1, F.3*F.6*F.9*F.12);;
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*F.6*F.7*F.9*F.11*F.12*F.13);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10*F.13);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10*F.11*F.13);;
gap> SetConjugate(coll, 5, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10*F.12);;
gap> SetConjugate(coll, 6, 1, F.7*F.13);;
gap> SetConjugate(coll, 6, 3, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.9*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.13);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.13);;
gap> SetConjugate(coll, 9, 2, F.9*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.13);;
gap> SetConjugate(coll, 10, 1, F.8*F.9*F.10*F.12);;
gap> SetConjugate(coll, 10, 2, F.10*F.12*F.13);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.8);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.13);;
gap> SetConjugate(coll, 13, 1, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);