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#72210409

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72209716 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#72209716C3
FrattiniC26C22⋊A4
socleC22#11709313
LCSG ▷ #72209716
UCSC1

38086 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 38086 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:

3,2^10|b^2=jk,c^2=j,d^2=fjk,e^2=fgij,b^a=bc,c^a=bj,d^a=eghj,d^b=df,d^c=dh,e^a=defghj,e^b=eg,e^c=efi,e^d=efghj,f^a=fgij,f^c=fj,f^e=fj,g^a=ghij,g^c=gjk,g^e=gjk,h^a=gj,h^b=hj,h^d=hj,h^e=hj,i^a=ij,i^b=ijk,i^c=ij,i^d=ijk,i^e=ik,j^a=jk,k^a=j

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC2×#72210665
maximal subgroup ofC2×#72210409
maximal normal subgroup of#72210665, C2×#72210409
maximal quotient of#72235031, #72237370, C2×#72210409

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