Finite Group Databasebeta

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#72219768

order18432
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23×C3

Aliases

No aliases for this group.

Splittings

Fitting#72213929 ⋊ C3⋊D3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#72213802C22
FrattiniC25C2×A4⋊S4
Fitting#72213929C3⋊D3
socleC24#11923523
hypocenterC42⋊C32D4
LCSG ▷ #72213802 ▷ C42⋊C32
UCSC1

4118 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 4118 conjugacy classes of subgroups — too many to draw. Only its 43 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:

2^3,3^2,2^8|a^2=c,f^2=j,g^2=k,h^2=l,i^2=m,b^a=bc,d^a=d^2,e^a=e^2,f^a=gj,f^b=fjk,f^c=fj,f^d=gk,f^e=fgj,g^a=fjk,g^b=gj,g^c=gk,g^d=fgk,g^e=fj,h^a=fijm,h^b=hm,h^c=hl,h^d=hi,h^e=fhk,i^a=ghjk,i^b=ilm,i^c=im,i^d=hm,i^e=fgik,j^a=k,j^d=k,j^e=jk,k^a=j,k^d=jk,k^e=j,l^a=jm,l^d=lm,l^e=jl,m^a=kl,m^d=l,m^e=jkm

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

holomorph ofC42⋊C3

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, 1, F.3);;
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, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.7*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.6*F.10);;
gap> SetConjugate(coll, 8, 1, F.6*F.9*F.10*F.13);;
gap> SetConjugate(coll, 8, 2, F.8*F.13);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.6*F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.12*F.13);;
gap> SetConjugate(coll, 9, 3, F.9*F.13);;
gap> SetConjugate(coll, 9, 4, F.8*F.13);;
gap> SetConjugate(coll, 9, 5, F.6*F.7*F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.11);;
gap> SetConjugate(coll, 10, 4, F.11);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> SetConjugate(coll, 11, 1, F.10);;
gap> SetConjugate(coll, 11, 4, F.10*F.11);;
gap> SetConjugate(coll, 11, 5, F.10);;
gap> SetConjugate(coll, 12, 1, F.10*F.13);;
gap> SetConjugate(coll, 12, 4, F.12*F.13);;
gap> SetConjugate(coll, 12, 5, F.10*F.12);;
gap> SetConjugate(coll, 13, 1, F.11*F.12);;
gap> SetConjugate(coll, 13, 4, F.12);;
gap> SetConjugate(coll, 13, 5, F.10*F.11*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);