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#72209707

order1536
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#346276 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#346276C3
FrattiniC25C22⋊A4
socleC22#73292
LCSG ▷ #346276
UCSC1

410 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 13 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:

3,2^9|b^2=h,c^2=fg,d^2=fj,e^2=fj,b^a=bc,c^a=bgi,c^b=cf,d^a=de,d^b=dfhj,d^c=dghi,e^a=d,e^b=eghij,e^c=egj,e^d=efj,f^a=fi,f^b=fi,f^c=fij,g^a=fhij,g^b=gj,g^c=gij,g^d=gj,g^e=gij,h^a=ghij,h^c=hi,h^d=hi,h^e=hj,i^a=j,j^a=ij

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72284333, #72284441, #72284484
maximal subgroup of#72235403, #72235523, #72284098, …
maximal normal subgroup of#72235403, #72235523, #72284098, …
maximal quotient of#72209708, #72210077, #72210079, …

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