Finite Group Databasebeta

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#72287287

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C27×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72287249 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4C22⋊#5464
commutator, Fitting, cosocle, hypocenter#72287249C3
FrattiniC22×C4C22⋊(C22⋊A4)
socleC2#72210049
LCSG ▷ #72287249
UCSC1 ◁ C4

10477 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 10477 conjugacy classes of subgroups — too many to draw. Only its 58 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:

3,2^10|b^2=k,c^2=k,d^2=ijk,e^2=ik,f^2=k,g^2=k,i^2=k,b^a=bc,c^a=b,c^b=ck,d^a=de,d^b=dj,d^c=di,e^a=dhjk,e^b=ei,e^c=eh,e^d=ehijk,f^a=fg,f^b=fk,f^c=fk,g^a=f,g^c=gk,g^f=gk,h^a=jk,h^b=hk,h^e=hk,i^a=ijk,i^b=ik,i^c=ik,i^d=ik,j^a=hj,j^c=jk,j^d=jk,j^e=jk

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72287613

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