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#72213093

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C25×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72213086 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC22C22⋊#5464
commutator, Fitting, cosocle, hypocenter#72213086C3
Frattini, socleC24C22⋊(C22⋊A4)
LCSG ▷ #72213086
UCSC1 ◁ C22

5853 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 5853 conjugacy classes of subgroups — too many to draw. Only its 72 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=h,c^2=hij,d^2=k,e^2=k,f^2=k,g^2=k,b^a=bc,c^a=bhj,c^b=ci,d^a=de,d^b=dh,d^c=dij,e^a=d,e^b=ej,e^c=ehij,e^d=ek,f^a=fg,f^b=fi,f^c=fi,f^d=fi,f^e=fik,g^a=f,g^c=gi,g^d=gk,g^e=gi,g^f=gk,h^a=j,j^a=hij

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72213221
maximal normal subgroup of#72213221
maximal quotient of#72267506, #72267621, #72267694

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