Finite Group Databasebeta

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#72272523

order3072
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C210

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72272519 ⋊ C3
derived#564 ⋊ C24×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#564C24×C3
Frattini, socleC22C24×C22⋊A4
Fitting#72272519C3
LCSG ▷ #564
UCSC1

At least 7491 conjugacy classes of subgroups; only the 537 normal subgroups are shown.

normal subgroups only

This group has at least 7491 conjugacy classes of subgroups — too many to draw. Only its 537 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:

2^4,3,2^6|f^2=k,g^2=j,h^2=j,i^2=k,e^b=ejk,e^c=ej,f^a=fk,f^b=fjk,f^d=fjk,f^e=fh,g^a=gj,g^b=gjk,g^c=gk,g^e=ij,g^f=gk,h^a=hj,h^b=hk,h^d=hk,h^e=fj,h^g=hjk,i^a=ik,i^b=ij,i^c=ijk,i^e=gi,i^f=ij,i^h=ik,j^e=k,k^e=jk

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272552, #72272624
maximal normal subgroup of#72272552, #72272624

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