Finite Group Databasebeta

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#72267937

order12288
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267906 ⋊ C3
derived#72267604 ⋊ C22×C3

Subgroups

subgroupG/H
center, socleC23#72209719
commutator, cosocle, hypocenter#72267604C22×C3
FrattiniC25×Q8C22×A4
Fitting#72267906C3
hypercenterC24#11709314
LCSG ▷ #72267604
UCSC1 ◁ C23 ◁ C24

At least 21039 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has at least 21039 conjugacy classes of subgroups — too many to draw. Only its 116 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|a^2=m,d^2=k,e^2=k,f^2=m,g^2=m,h^2=m,i^2=m,d^a=dfgm,d^b=dfgi,d^c=e,e^a=ef,e^b=efhm,e^c=dek,e^d=ek,f^a=fm,f^c=gj,f^d=fj,g^c=fgjm,g^d=gj,g^e=gj,g^f=gm,h^a=hm,h^c=hijkl,h^d=hjkl,h^g=hm,i^a=im,i^c=h,i^e=ijkl,i^f=im,i^g=im,i^h=im,j^a=jm,k^a=km,l^a=lm

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267998

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