Finite Group Databasebeta

← Prev (#72257979) Next (#72257981) →

#72257980

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C27×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, SylowC22×#31076 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4C22⋊#5464
commutator, Fitting, cosocle, hypocenterC22×#31076C3
FrattiniC22×C4C22⋊(C22⋊A4)
socleC23#2679⋊C3
LCSG ▷ C22×#31076
UCSC1 ◁ C4

10828 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 10828 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 48

Polycyclic presentation:

3,2^10|b^2=h,c^2=hi,j^2=k,b^a=bc,c^a=bhi,d^a=de,d^b=dh,d^c=dhij,e^a=d,e^b=ehij,e^c=ehik,f^a=fg,f^b=fhij,f^c=fhik,f^d=fk,g^a=f,g^b=gijk,g^c=gj,g^d=gk,g^e=gk,h^a=i,h^e=hk,h^f=hk,h^g=hk,i^a=hi,i^d=ik,i^e=ik,i^f=ik,j^a=hij,j^e=jk,j^f=jk,j^g=jk

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72260019, #72260812, #72267242
maximal normal subgroup of#72267242
maximal quotient of#72260019, #72267247

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