Finite Group Databasebeta

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#72258960

order18432
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C25×C3

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72258915 ⋊ C32
derivedC44 ⋊ C23×C32

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC44C23×C32
Frattini, socleC24C23×A42
Fitting#72258915C32
LCSG ▷ C44
UCSC1

14462 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 14462 conjugacy classes of subgroups — too many to draw. Only its 189 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) = 96

Polycyclic presentation:

2^3,3^2,2^8|f^2=j,g^2=k,h^2=l,i^2=m,f^a=fj,f^b=fk,f^c=fj,f^d=fg,f^e=gjk,g^a=gk,g^b=gjk,g^c=gk,g^d=fk,g^e=fgj,h^a=hjk,h^b=hlm,h^c=hl,h^d=il,h^e=ghjk,i^a=ik,i^b=il,i^c=im,i^d=hi,i^e=fik,j^d=jk,j^e=k,k^d=j,k^e=jk,l^d=m,l^e=kl,m^d=lm,m^e=jm

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

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", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.12);;
gap> SetPower(coll, 9, F.13);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.12*F.13);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.9*F.12);;
gap> SetConjugate(coll, 8, 5, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.13);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.6*F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 11, 4, F.10);;
gap> SetConjugate(coll, 11, 5, F.10*F.11);;
gap> SetConjugate(coll, 12, 4, F.13);;
gap> SetConjugate(coll, 12, 5, F.11*F.12);;
gap> SetConjugate(coll, 13, 4, F.12*F.13);;
gap> SetConjugate(coll, 13, 5, F.10*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);