Finite Group Databasebeta

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#72213620

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72213602 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#72213602C3
Frattini#47563A4
socleC22#7318⋊C3
LCSG ▷ #72213602
UCSC1

260 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 9 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=d,c^2=e,d^2=j,f^2=i,b^a=bc,c^a=befgk,c^b=cf,d^a=defghik,d^c=dg,e^a=dh,e^b=egh,e^d=ei,f^a=fgi,f^b=fgi,f^c=fghik,f^e=fj,g^a=hijk,g^b=gj,g^c=gi,g^e=gk,g^f=gj,h^a=ghijk,h^b=hij,h^c=hijk,h^d=hj,h^e=hk,h^f=hjk,i^a=ij,i^b=ij,i^c=ik,j^a=jk,k^a=j

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72213625, #72267658, #72267659

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