Finite Group Databasebeta

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#72340821

order6144
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C2×#72224760
commutator, cosocle#59816⋊C3C23
FrattiniC22×C4C22×#1020
Fitting#72236189D3
socleC23C2×#87524
prime radical#72237738C2
hypercenterC4#72224760
hypocenter#2679⋊C3C2×D4
LCSG ▷ #59816⋊C3 ▷ #2679⋊C3
UCSC1 ◁ C2 ◁ C4

15146 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 15146 conjugacy classes of subgroups — too many to draw. Only its 65 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:

2^4,3,2^7|b^2=djkl,d^2=j,f^2=k,g^2=kl,b^a=bd,c^b=cjkl,d^a=dj,e^a=ejkl,e^b=e^2,e^c=el,e^d=ejk,f^a=fjkl,f^b=fg,f^c=fjk,f^d=fjk,f^e=gjk,g^a=gl,g^b=gjl,g^c=gjkl,g^d=gjkl,g^e=fgj,g^f=gj,h^a=hjkl,h^b=fgikl,h^c=hjkl,h^d=hjkl,h^e=hi,h^f=hk,h^g=hjl,i^a=ijk,i^b=fh,i^c=ijk,i^d=ijk,i^e=h,i^f=ikl,i^g=ijk,k^b=jl,k^e=kl,l^b=jk,l^e=jk

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340866
maximal normal subgroup of#72340866

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