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#72209692

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72209689 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC42#5464
commutator, Fitting, cosocle, hypocenter#72209689C3
FrattiniC22×C42C22⋊A4
socleC22#11709313
LCSG ▷ #72209689
UCSC1 ◁ C42

1428 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

3,2^10|d^2=jk,e^2=jk,f^2=j,g^2=k,h^2=jk,i^2=j,b^a=c,c^a=bc,d^a=e,d^b=dg,d^c=dgij,e^a=dejk,e^b=ef,e^c=efhjk,e^d=ejk,f^a=fghijk,f^b=fj,f^c=fjk,f^d=fjk,f^e=fj,g^a=fhjk,g^b=gk,g^c=gj,g^d=gk,g^e=gjk,h^a=fgk,h^b=hk,h^c=hj,h^d=hk,h^e=hjk,i^a=fk,i^b=ijk,i^c=ik,i^d=ij,i^e=ik

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209693
maximal quotient of#72209979

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