Finite Group Databasebeta

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#72261057

order12288
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#72260850 ⋊ D3
derived#72260911 ⋊ C22

Subgroups

subgroupG/H
centerC2#72260581
commutator, cosocle, hypocenter#72260911C22
FrattiniC24C2×#87588
Fitting#72260850D3
socleC23C2×#11715850
hypercenterC22#72260462
LCSG ▷ #72260911
UCSC1 ◁ C2 ◁ C22

At least 12343 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has at least 12343 conjugacy classes of subgroups — too many to draw. Only its 47 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:

2^2,3,2^10|a^2=m,b^2=m,f^2=k,g^2=j,b^a=bm,c^a=c^2,c^b=cjm,d^a=de,d^c=e,e^c=de,f^a=gk,f^b=fkm,f^c=gkm,g^a=fj,g^b=gjm,g^c=fgm,g^f=gm,h^c=i,h^d=hjkl,h^e=hjkm,i^a=hi,i^c=hi,i^d=ikl,i^e=il,j^a=k,j^c=jkm,k^a=j,k^c=j,l^a=jklm,l^c=klm

Conjugacy classes & character table

92 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", "m");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.13);;
gap> SetPower(coll, 2, F.13);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.13);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.10*F.13);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.7*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.11*F.13);;
gap> SetConjugate(coll, 6, 3, F.7*F.11*F.13);;
gap> SetConjugate(coll, 7, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.13);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.13);;
gap> SetConjugate(coll, 7, 6, F.7*F.13);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.10*F.11*F.13);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11*F.13);;
gap> SetConjugate(coll, 11, 1, F.10);;
gap> SetConjugate(coll, 11, 3, F.10);;
gap> SetConjugate(coll, 12, 1, F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 12, 3, F.11*F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);