Finite Group Databasebeta

← Prev (#72261697) Next (#72261699) →

#72261698

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2#72260915
commutator, cosocle#72240348C22
FrattiniC26C2×#1020
Fitting#72233929D3
socleC25#87524
prime radical#72260766C2
hypocenter#72240347D4
LCSG ▷ #72240348 ▷ #72240347
UCSC1 ◁ C2

At least 16840 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has at least 16840 conjugacy classes of subgroups — too many to draw. Only its 48 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:

2^3,3,2^9|a^2=k,b^2=c,e^2=i,f^2=j,g^2=ilm,h^2=jl,b^a=bc,d^b=d^2,e^a=eik,e^b=ei,e^d=fik,f^a=fjk,f^b=ef,f^d=efk,f^e=fk,g^a=giklm,g^b=efhm,g^c=gl,g^d=hiklm,g^e=gm,g^f=gkl,h^a=hjkl,h^b=fgj,h^c=hm,h^d=ghk,h^e=hkl,h^f=hlm,h^g=hk,i^d=j,j^b=ijk,j^d=ijk,l^b=m,l^d=m,m^b=l,m^d=lm

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

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