Finite Group Databasebeta

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#72249582

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72249531 ⋊ C3

Subgroups

subgroupG/H
center, socle, hypercenterC22#72209702
commutator, Fitting, cosocle, hypocenter#72249531C3
FrattiniC26C22⋊(C22⋊A4)
LCSG ▷ #72249531
UCSC1 ◁ C22

44728 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 44728 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) = 256

Polycyclic presentation:

3,2^12|b^2=k,c^2=j,d^2=jl,e^2=jkl,f^2=l,g^2=l,b^a=ck,c^a=bc,d^a=ej,d^b=djkm,d^c=djm,e^a=del,e^b=ej,e^c=ekm,e^d=el,f^a=g,f^b=fi,f^c=fh,f^d=fhilm,f^e=fi,g^a=fgl,g^b=ghm,g^c=ghim,g^d=gilm,g^e=ghl,g^f=gl,h^a=im,h^b=hl,h^c=hm,h^d=hl,h^e=hm,i^a=hilm,i^b=ilm,i^c=il,i^d=ilm,i^e=il,j^a=jk,j^f=jm,j^g=jlm,k^a=j,k^f=klm,k^g=kl

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72249859

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