Finite Group Databasebeta

← Prev (#72271331) Next (#72271333) →

#72271332

order12288
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268229 ⋊ C3
derived#72268220 ⋊ C22×C3

Subgroups

subgroupG/H
center, socleC22#72209720
commutator, cosocle, hypocenter#72268220C22×C3
FrattiniC22×#543C22×A4
Fitting#72268229C3
hypercenterC24#11709314
LCSG ▷ #72268220
UCSC1 ◁ C22 ◁ C24

At least 14708 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has at least 14708 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^10|d^2=j,e^2=j,h^2=lm,i^2=lm,d^a=dg,d^b=dh,d^c=e,e^a=ef,e^b=ei,e^c=dej,e^d=ej,f^b=fl,f^c=fgj,f^d=fj,g^b=gl,g^c=f,g^e=gj,h^a=hl,h^b=hlm,h^c=i,h^d=hm,h^e=hjk,h^f=hl,i^a=il,i^b=ilm,i^c=hijkl,i^d=ijkm,i^e=im,i^g=il,i^h=ilm,j^b=jl,k^a=kl,k^b=km

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271469

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