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#72209697

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72209694 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC22×C4#5464
commutator, Fitting, cosocle, hypocenter#72209694C3
FrattiniC24×C4C22⋊A4
socleC23#2679⋊C3
LCSG ▷ #72209694
UCSC1 ◁ C22×C4

4485 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 4485 conjugacy classes of subgroups — too many to draw. Only its 51 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|b^2=f,c^2=fh,d^2=fgij,e^2=fij,f^2=k,j^2=k,b^a=cg,c^a=bcfg,c^b=cfg,d^a=eghk,d^b=dj,d^c=dhi,e^a=defhijk,e^b=eijk,e^c=eghjk,e^d=efhijk,f^a=fhk,f^c=fk,g^a=ghk,g^c=gk,h^a=gk,h^b=hk,h^c=hk,i^a=hik,i^c=ik,j^a=ghjk,j^b=jk,j^c=jk

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209698
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, 2, F.6);;
gap> SetPower(coll, 3, F.6*F.8);;
gap> SetPower(coll, 4, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.6*F.9*F.10);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.5*F.7*F.8*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);