Finite Group Databasebeta

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#72241454

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C25×C44

Aliases

No aliases for this group.

Splittings

Fitting, derived, SylowUT(4,𝔽4) ⋊ C3

Subgroups

subgroupG/H
center, socle, hypercenterC22#72209702
commutator, Fitting, cosocle, hypocenterUT(4,𝔽4)C3
FrattiniC26C22⋊(C22⋊A4)
LCSG ▷ UT(4,𝔽4)
UCSC1 ◁ C22

55363 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 55363 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 64

Polycyclic presentation:

3,2^12|d^2=m,e^2=m,f^2=l,g^2=l,b^a=c,c^a=bc,d^a=e,d^b=dk,d^c=dj,e^a=dem,e^b=ej,e^c=ejkm,e^d=em,f^a=g,f^b=fi,f^c=fh,f^d=fl,f^e=fm,g^a=fgl,g^b=ghl,g^c=ghi,g^d=glm,g^e=gl,g^f=gl,h^a=il,h^d=hl,h^e=hlm,i^a=hi,i^d=im,i^e=il,j^a=km,j^f=jl,j^g=jlm,k^a=jkm,k^f=km,k^g=kl

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72242443

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