Finite Group Databasebeta

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#72209773

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#8093108 ⋊ D3
derived#72209768 ⋊ C2

Subgroups

subgroupG/H
center, socleC22#11715850
commutator, cosocle, hypocenter#72209768C2
Frattini, hypercenterC23#87588
Fitting#8093108D3
LCSG ▷ #72209768
UCSC1 ◁ C22 ◁ C23

4832 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 4832 conjugacy classes of subgroups — too many to draw. Only its 29 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:

2,3,2^9|a^2=k,c^2=i,d^2=i,g^2=ik,h^2=ik,b^a=b^2,c^a=gik,c^b=d,d^a=gh,d^b=cdi,d^c=di,e^a=f,e^b=f,e^c=ej,e^d=eij,f^a=e,f^b=ef,f^c=fi,f^d=fj,g^a=ci,g^b=h,g^c=gk,g^e=gik,g^f=gjk,h^a=cd,h^b=ghik,h^c=hk,h^d=hk,h^e=hij,h^f=hik,h^g=hik,i^a=ik,j^a=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209775, #72227742, #72227824
maximal normal subgroup of#72227742, #72227824
maximal quotient of#72227611, #72227636, #72258531

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