Finite Group Databasebeta

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#72258563

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

Aliases

No aliases for this group.

Splittings

Fitting#72222661 ⋊ D3
derived#72222569 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#72219737
commutator, cosocle, hypocenter#72222569C22
Frattini, socleC25C2×#1020
Fitting#72222661D3
LCSG ▷ #72222569
UCSC1 ◁ C2

11053 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 11053 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 48

Polycyclic presentation:

2^2,3,2^9|d^2=h,e^2=i,f^2=j,g^2=k,c^a=c^2,d^a=eh,d^b=dhl,d^c=ehl,e^a=di,e^b=eil,e^c=del,e^d=el,f^a=dgijk,f^b=fj,f^c=fg,f^d=fl,f^e=fl,g^a=efhjk,g^b=gk,g^c=fk,g^d=gl,h^a=i,h^c=i,i^a=h,i^c=hil,j^a=hkl,j^c=jk,k^a=ijl,k^c=j

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

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