Finite Group Databasebeta

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#72218289

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#233527 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72218248
commutator, cosocle, hypocenter#1273⋊C3C23
FrattiniC2×C42C22×S4
Fitting#233527D3
socleC23C2×#857⋊C2
LCSG ▷ #1273⋊C3
UCSC1 ◁ C2

1342 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 32 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:

2^3,3,2^7|a^2=k,b^2=i,c^2=jk,e^2=g,f^2=h,g^2=j,h^2=k,b^a=bik,c^a=chjk,c^b=cjk,d^a=d^2j,d^b=djk,d^c=dhi,e^a=efgijk,e^b=egij,e^c=ej,e^d=fgijk,f^a=fi,f^b=fhik,f^c=fk,f^d=efi,f^e=fi,g^a=ghij,g^b=gj,g^d=hj,h^b=hk,h^d=ghi,j^a=jk,j^d=k,k^d=jk

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72218314, #72218337, #72218356

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