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#72209802

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11714279
commutator, prime radical, cosocle, hypocenter#72209796C2
FrattiniC23×C4#1020
Fitting#475912D3
hypercenterC2×C4#85559
LCSG ▷ #72209796
UCSC1 ◁ C22 ◁ C2×C4

701 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 701 conjugacy classes of subgroups — too many to draw. Only its 20 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:

2,3,2^9|a^2=ij,c^2=g,d^2=gh,e^2=hjk,f^2=j,i^2=k,j^2=k,b^a=b^2,c^a=dg,c^b=dg,d^a=cgh,d^b=cd,e^a=dfhijk,e^b=fg,e^c=eg,e^d=egik,f^a=ceij,f^b=efghjk,f^c=fgi,f^d=fghk,f^e=fghjk,g^a=gh,g^b=gh,g^f=gk,h^b=g,h^e=hk,h^f=hk,i^a=hik,i^b=gi,i^e=ik,j^a=hjk,j^b=gj,j^e=jk

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209804
maximal quotient of#72227666

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