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#72209708

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C2×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72209704 ⋊ C3

Subgroups

subgroupG/H
center, socle, hypercenterC2#72209707
commutator, Fitting, cosocle, hypocenter#72209704C3
FrattiniC23×Q8C22⋊A4
LCSG ▷ #72209704
UCSC1 ◁ C2

485 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 14 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=g,c^2=fgh,d^2=fijk,e^2=fijk,g^2=k,h^2=k,b^a=cfgik,c^a=bcf,c^b=cf,d^a=e,d^b=dfgijk,d^c=dhj,e^a=defijk,e^b=ehik,e^c=eghijk,e^d=efijk,f^a=fi,f^b=fj,f^c=fi,g^a=fghijk,g^c=gj,g^d=gj,g^e=gij,h^a=gk,h^b=hij,h^c=hij,h^d=hi,h^e=hj,h^g=hk,i^a=ij,i^b=ik,j^a=i,j^c=jk

Conjugacy classes & character table

23 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209709
maximal quotient of#72210078, #72210080, #72210092

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