Finite Group Databasebeta

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#72271440

order6144
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72271430 ⋊ C3
derived#72268378 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC23#11710337
commutator, cosocle, hypocenter#72268378C2×C3
FrattiniC23×Q8C2×C22⋊A4
Fitting#72271430C3
hypercenterC24#85645
LCSG ▷ #72268378
UCSC1 ◁ C23 ◁ C24

7269 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 7269 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) = 256

Polycyclic presentation:

2,3,2^10|a^2=l,c^2=i,d^2=i,e^2=kl,f^2=kl,g^2=l,h^2=l,c^a=cg,c^b=cd,d^a=dghl,d^b=c,d^c=di,e^b=ef,e^c=ej,e^d=ejk,f^b=e,f^c=fk,f^d=fj,f^e=fkl,g^a=gl,g^b=hil,g^d=gil,g^e=gl,h^b=ghi,h^c=hil,h^d=hil,h^f=hl,h^g=hl,i^a=il,j^a=jl

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271486
maximal normal subgroup of#72271486

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