Finite Group Databasebeta

← Prev (#72267768) Next (#72267770) →

#72267769

order12288
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267766 ⋊ C3
derived#72267764 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2#72226626
commutator, cosocle, hypocenter#72267764C2×C3
FrattiniC2×#7555C2×A4
Fitting#72267766C3
socleC23#72226625
LCSG ▷ #72267764
UCSC1 ◁ C2

11229 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 11229 conjugacy classes of subgroups — too many to draw. Only its 27 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:

2,3,2^11|c^2=g,d^2=ghi,e^2=ik,f^2=hkl,b^a=bim,c^a=cf,c^b=cd,d^a=de,d^b=cim,d^c=dgh,e^a=eik,e^b=fl,e^c=egj,e^d=eik,f^a=fhkl,f^b=efgj,f^c=fhkl,f^d=fgjkl,f^e=fk,g^b=gilm,g^d=gm,h^b=hil,h^c=hm,h^d=hl,i^b=hlm,i^c=il,i^d=ilm,j^a=jkl,j^b=hijl,j^d=jl,j^e=jlm,j^f=jm,k^b=km,k^c=km,k^d=klm,l^b=lm,m^b=l

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267770

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