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#72272647

order3072
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C210

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72272580 ⋊ C3
derivedC22×H(3,𝔽4) ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×H(3,𝔽4)C22×C3
Frattini, socleC22C22×C22⋊(C22⋊A4)
Fitting#72272580C3
LCSG ▷ C22×H(3,𝔽4)
UCSC1

At least 25314 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has at least 25314 conjugacy classes of subgroups — too many to draw. Only its 226 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:

2^2,3,2^8|c^b=cjk,d^a=djk,d^b=dk,d^c=e,e^a=ej,e^b=ejk,e^c=de,f^b=fk,f^c=defg,g^a=gk,g^c=df,h^a=hk,h^b=hjk,h^c=i,h^d=hj,h^e=hjk,h^f=hk,h^g=hj,i^a=ijk,i^b=ij,i^c=hi,i^d=ijk,i^e=ik,i^f=ij,i^g=ijk,j^c=k,k^c=jk

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272663
maximal normal subgroup of#72272663

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