Finite Group Databasebeta

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#72271387

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72271377 ⋊ C3
derived#72268378 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#72209717
commutator, cosocle, hypocenter#72268378C2×C3
FrattiniC23×Q8C2×C22⋊A4
Fitting#72271377C3
hypercenterC24#85645
LCSG ▷ #72268378
UCSC1 ◁ C22 ◁ C24

6676 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 6676 conjugacy classes of subgroups — too many to draw. Only its 42 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:

2,3,2^10|a^2=kl,c^2=i,d^2=i,e^2=k,f^2=k,g^2=kl,h^2=kl,c^a=ch,c^b=cd,d^a=dg,d^b=c,d^c=di,e^b=ef,e^c=eik,f^b=e,f^c=fik,f^d=fik,f^e=fk,g^a=gkl,g^b=h,g^c=gj,g^d=gl,g^f=gk,h^a=hkl,h^b=ghjk,h^c=hl,h^d=hjl,h^e=hk,h^f=hk,h^g=hkl,i^a=ik,j^a=jkl

Conjugacy classes & character table

70 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.11*F.12);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.11*F.12);;
gap> SetPower(coll, 8, F.11*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
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.9*F.11);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11*F.12);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.10*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);