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#72258000

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C28

Aliases

No aliases for this group.

Splittings

FittingC22×H(7,𝔽2) ⋊ D3

Subgroups

subgroupG/H
center, FrattiniC2C22×#87588
commutator, cosocle, hypocenterC22⋊#997C23
FittingC22×H(7,𝔽2)D3
socleC23C22×#1020
hypercenterD4#87588
LCSG ▷ C22⋊#997
UCSC1 ◁ C2 ◁ D4

At least 9946 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has at least 9946 conjugacy classes of subgroups — too many to draw. Only its 98 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|e^2=k,f^2=k,g^2=k,h^2=k,b^a=bk,d^c=d^2,e^c=fk,e^d=f,f^c=ek,f^d=efk,f^e=fk,g^c=egh,g^d=h,g^f=gk,h^c=efh,h^d=ghk,h^e=hk,h^g=hk,i^c=ejk,i^d=ij,i^e=ik,i^f=ik,i^g=ik,i^h=ik,j^c=fik,j^d=i,j^e=jk,j^g=jk

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72312045
maximal normal subgroup of#72312045

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