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#72209792

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Splittings

Fitting#417744 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72209644
commutator, cosocle, hypocenter#2273⋊C3C23
FrattiniC25C22×S4
Fitting#417744D3
socleC23C2×#1020⋊C2
LCSG ▷ #2273⋊C3
UCSC1 ◁ C2

5229 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 5229 conjugacy classes of subgroups — too many to draw. Only its 37 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|c^2=ij,e^2=i,f^2=j,c^a=cij,c^b=ck,d^a=dik,d^c=d^2jk,e^a=eik,e^b=eghk,e^c=fgik,e^d=fik,f^a=fjk,f^b=fgk,f^c=eghjk,f^d=efk,f^e=fk,g^c=gh,g^d=h,g^e=gk,h^d=gh,h^e=hk,h^f=hk,i^c=j,i^d=j,j^c=i,j^d=ijk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209793
maximal quotient of#72227741

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