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#72210079

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C2×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72210067 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72209707
commutator, Fitting, cosocle, hypocenter#72210067C3
FrattiniC26C22⋊A4
socleC23#73292
LCSG ▷ #72210067
UCSC1 ◁ C2

885 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 21 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=g,d^2=fghk,e^2=fghk,b^a=bc,c^a=bgij,c^b=ci,d^a=de,d^b=dh,d^c=dgik,e^a=d,e^b=efhijk,e^c=eghi,e^d=efghk,f^a=fgik,f^c=fj,f^d=fk,f^e=fjk,g^a=fjk,g^b=gk,g^d=gk,g^e=gjk,h^a=fhik,h^b=hk,h^c=hj,i^a=ij,i^b=ij,i^c=ik,i^d=ik,i^e=ijk,j^a=jk,k^a=j

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72210080, #72266485

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