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#72216213

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#72209716 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#72215614C22
FrattiniC22×#120C2×S4
Fitting#72209716D3
socleC22#72215875
hypocenterC22⋊(C22⋊(C22⋊A4))D4
LCSG ▷ #72215614 ▷ C22⋊(C22⋊(C22⋊A4))
UCSC1

22642 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 22642 conjugacy classes of subgroups — too many to draw. Only its 18 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:

2^3,3,2^8|b^2=c,b^a=bc,d^a=d^2,d^b=d^2,e^b=egi,e^c=eghkl,e^d=f,f^a=ef,f^b=efghij,f^c=fgk,f^d=ef,g^a=h,g^b=hl,g^d=h,h^a=g,h^b=gk,h^d=gh,i^b=ik,i^c=ikl,i^d=j,j^a=ij,j^b=ijkl,j^c=jk,j^d=ij,k^a=l,k^b=l,k^d=l,l^a=k,l^b=k,l^d=kl

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72216213
maximal normal subgroup ofC2×#72216213
maximal quotient ofC2×#72216213
automorphism group of#543

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