Finite Group Databasebeta

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#72258797

order12288
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

Fitting#72258564 ⋊ D3
derived#72258498 ⋊ C22

Subgroups

subgroupG/H
centerC2#72258322
commutator, cosocle, hypocenter#72258498C22
FrattiniC26C2×#1020
Fitting#72258564D3
socleC23#72209653
hypercenterC22#72216034
LCSG ▷ #72258498
UCSC1 ◁ C2 ◁ C22

At least 34397 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has at least 34397 conjugacy classes of subgroups — too many to draw. Only its 34 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|c^a=c^2,d^a=e,d^b=dh,d^c=e,e^a=d,e^b=ei,e^c=de,f^b=fj,f^c=g,f^d=fl,f^e=fl,g^a=fg,g^b=gk,g^c=fg,g^e=gl,h^a=i,h^c=i,h^f=hm,i^a=h,i^c=hi,i^f=im,i^g=im,j^c=k,k^a=jk,k^c=jk,l^b=lm

Conjugacy classes & character table

88 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> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 1, F.4);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, 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.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.13);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.13);;
gap> SetConjugate(coll, 9, 7, F.9*F.13);;
gap> SetConjugate(coll, 10, 3, F.11);;
gap> SetConjugate(coll, 11, 1, F.10*F.11);;
gap> SetConjugate(coll, 11, 3, F.10*F.11);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);