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#72219764

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Splittings

Fitting#72213929 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#72210013C22
FrattiniC25C2×#1020
Fitting#72213929D3
socleC24#87524
hypocenter#11709197D4
LCSG ▷ #72210013 ▷ #11709197
UCSC1

5299 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 5299 conjugacy classes of subgroups — too many to draw. Only its 29 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:

2^3,3,2^8|a^2=c,e^2=i,f^2=j,g^2=k,h^2=l,b^a=bc,d^a=d^2,e^a=fj,e^b=eij,e^c=ei,e^d=ef,f^a=e,f^b=fi,f^c=fj,f^d=ej,g^a=ehl,g^b=gl,g^c=gk,g^d=efghj,h^a=fgj,h^b=hkl,h^c=hl,h^d=fgijl,i^a=j,i^d=ij,j^a=i,j^d=i,k^a=il,k^d=ijkl,l^a=jk,l^d=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72219768

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