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#72210808

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72210224
commutator, prime radical, cosocle, hypocenter#72210255C2
Frattini#1726S4
Fitting#93255D3
hypercenterC8#67992
LCSG ▷ #72210255
UCSC1 ◁ C2 ◁ C4 ◁ C8

182 conjugacy classes of subgroups; only the 12 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 12 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:

2,3,2^9|a^2=hj,c^2=e,d^2=f,e^2=h,f^2=i,g^2=jk,h^2=k,i^2=k,j^2=k,b^a=b^2hi,c^a=cgk,c^b=deghijk,d^a=cdefhij,d^b=cdg,d^c=dg,e^a=ejk,e^b=fh,e^d=ej,f^a=efghi,f^b=efgjk,f^c=fjk,f^e=fk,g^a=gj,h^a=hk,h^b=ik,h^d=hk,i^a=hij,i^b=hijk,i^c=ik,j^a=jk

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72210811, #72210816

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