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#72213294

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2#72213198
commutator, cosocle, hypocenter#7014⋊C3C22
FrattiniC43C2×S4
Fitting#102857D3
socleC23#72997
prime radical#72211034C2
hypercenterC4#11711452
LCSG ▷ #7014⋊C3
UCSC1 ◁ C2 ◁ C4

541 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 16 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:

2^2,3,2^8|a^2=j,b^2=h,d^2=f,e^2=g,f^2=i,g^2=j,h^2=k,b^a=bhk,c^a=c^2ik,c^b=cik,d^a=defhijk,d^b=dfhik,d^c=efhijk,e^a=ehk,e^b=eghj,e^c=deh,e^d=eh,f^a=fghi,f^b=fik,f^c=gi,f^e=fk,g^a=gk,g^b=gjk,g^c=fghk,g^d=gk,h^a=hk,i^a=ijk,i^c=j,j^c=ijk

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214044
maximal normal subgroup of#72214044
maximal quotient of#72213299

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