Finite Group Databasebeta

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#72243519

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72242720 ⋊ C3

Subgroups

subgroupG/H
center, socle, hypercenterC22#72209702
commutator, Fitting, cosocle, hypocenter#72242720C3
FrattiniC26C22⋊(C22⋊A4)
LCSG ▷ #72242720
UCSC1 ◁ C22

55149 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 55149 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|b^2=h,c^2=hi,b^a=chlm,c^a=bclm,c^b=clm,d^a=e,d^b=dj,d^c=djk,e^a=de,e^b=ejk,e^c=ekm,f^a=g,f^b=fhikl,f^c=fhjm,f^d=fl,f^e=fl,g^a=fg,g^b=ghjm,g^c=gijkm,g^e=gl,h^a=hi,h^d=hm,h^e=hl,h^f=hlm,h^g=hm,i^a=hlm,i^d=il,i^e=ilm,i^f=im,i^g=il,j^a=klm,j^b=jm,j^c=jl,j^f=jlm,j^g=jm,k^a=jkm,k^b=klm,k^c=km,k^f=kl,k^g=klm

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72244262

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