Finite Group Databasebeta

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#72257974

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C27×C43

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72252252 ⋊ C3

Subgroups

subgroupG/H
center, socle, hypercenterC22#72209702
commutator, Fitting, cosocle, hypocenter#72252252C3
FrattiniC26C22⋊(C22⋊A4)
LCSG ▷ #72252252
UCSC1 ◁ C22

Only the 86 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 86 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:

3,2^12|b^2=h,c^2=i,d^2=ijl,e^2=hikl,b^a=chm,c^a=bcm,c^b=cm,d^a=eik,d^b=dj,d^c=djl,e^a=dejkl,e^b=ejlm,e^c=ejk,e^d=ejkl,f^a=g,f^b=fhjm,f^c=fhikm,f^d=fhikm,f^e=fijk,g^a=fg,g^b=ghik,g^c=gijk,g^d=gijkm,g^e=ghjm,h^a=i,h^d=hm,h^f=hm,i^a=him,i^d=im,i^e=im,i^f=im,i^g=im,j^a=jkm,j^b=jm,j^e=jm,j^f=jm,k^a=jm,k^c=km,k^d=km,k^e=km,k^g=km,l^a=jl,l^b=lm,l^c=lm,l^d=lm,l^f=lm,l^g=lm

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72257978

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