Finite Group Databasebeta

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#72218323

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

Aliases

No aliases for this group.

Splittings

Fitting#72211642 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC22#72218248
commutator, hypocenter#1273⋊C3C22×C4
FrattiniC22×C42C22×S4
Fitting#72211642D3
socleC24C2×#857⋊C2
prime radicalC2×#72213204C2
cosocleC2×#1273⋊C3C23
LCSG ▷ #1273⋊C3
UCSC1 ◁ C22

3511 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 3511 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^7|a^2=jl,b^2=j,c^2=d,f^2=h,g^2=i,h^2=k,i^2=l,b^a=bl,c^a=cikl,c^b=ckl,d^a=dl,e^a=e^2k,e^b=ekl,e^c=eij,e^d=el,f^a=fghjkl,f^b=fhjk,f^c=fk,f^e=ghjkl,g^a=gj,g^b=gijl,g^c=gl,g^e=fgj,g^f=gj,h^a=hijk,h^b=hk,h^e=ik,i^b=il,i^e=hij,k^a=kl,k^e=l,l^e=kl

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72218420

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