Finite Group Databasebeta

← Prev (#72267023) Next (#72267025) →

#72267024

order12288
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267008 ⋊ C3
derived#72267006 ⋊ C2×C3

Subgroups

subgroupG/H
centerC22#72209714
commutator, cosocle, hypocenter#72267006C2×C3
FrattiniC24×Q8C2×C22⋊A4
Fitting#72267008C3
socleC24#11710337
hypercenterC23#72209713
LCSG ▷ #72267006
UCSC1 ◁ C22 ◁ C23

Only the 53 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 53 normal subgroups are shown; non-normal subgroups are omitted.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,2^11|c^2=i,d^2=ij,e^2=il,f^2=ijkl,g^2=m,h^2=m,c^a=ch,c^b=cd,d^a=dg,d^b=cjk,d^c=dik,e^a=egm,e^b=ef,e^c=el,e^d=ejlm,f^a=fghijl,f^b=ek,f^c=fkm,f^d=fkl,f^e=fijl,g^a=gm,g^b=h,g^c=gijl,g^f=gijlm,h^a=hm,h^b=ghijlm,h^d=hijl,h^e=hijlm,h^f=hijlm,h^g=hm,i^a=im,i^b=ijk,j^b=jk,k^b=j,l^b=jl

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267138

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