Finite Group Databasebeta

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#72267891

order6144
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72267889 ⋊ C3
derived#7642105 ⋊ C22×C3

Subgroups

subgroupG/H
center, socleC22#72209719
commutator, cosocle, hypocenter#7642105C22×C3
FrattiniC24×Q8C22×A4
Fitting#72267889C3
hypercenterC23#11709314
LCSG ▷ #7642105
UCSC1 ◁ C22 ◁ C23

39606 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 39606 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 128

Polycyclic presentation:

2^2,3,2^9|a^2=l,d^2=k,e^2=k,f^2=l,g^2=l,h^2=l,i^2=l,d^a=dg,d^b=dghl,d^c=de,e^a=ef,e^b=efhi,e^c=d,e^d=ek,f^a=fl,f^c=g,f^d=fj,g^a=gl,g^c=fgjl,g^e=gj,g^f=gl,h^a=hl,h^c=ijl,h^e=hjl,h^f=hl,i^c=hij,i^d=ijl,i^e=ijl,i^f=il,i^g=il,i^h=il,j^a=jl,k^a=kl

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267937

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