Finite Group Databasebeta

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#72209763

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC23#87588
commutator, prime radical, cosocle, hypocenter#72209758C2
Fitting#7641846D3
LCSG ▷ #72209758
UCSC1 ◁ C23

4835 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 4835 conjugacy classes of subgroups — too many to draw. Only its 46 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=j,c^2=j,d^2=j,e^2=ij,f^2=ij,g^2=ijk,h^2=ijk,b^a=b^2,c^a=dj,c^b=d,d^a=cj,d^b=cdj,d^c=dj,e^a=defgijk,e^b=f,e^c=ei,e^d=eijk,f^a=cfghij,f^b=efij,f^c=fjk,f^d=fi,f^e=fij,g^a=dhj,g^b=h,g^c=gj,g^d=gijk,g^e=gjk,g^f=gj,h^a=cgj,h^b=ghijk,h^c=hik,h^d=hj,h^e=hk,h^f=hjk,h^g=hijk

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209765, #72227751, #72227869
maximal normal subgroup of#72227751, #72227869
maximal quotient of#72227611

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