Finite Group Databasebeta

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#72226640

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72226638 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#72226638C3
FrattiniC26C22⋊A4
socleC22#11709313
LCSG ▷ #72226638
UCSC1

1889 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 1889 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(2,8,23)(3,15,14)(5,6,7)(9,11,17)(10,25,29)(12,19,24)(13,22,28)(16,21,20)(18,30,31)(26,27,32);(2,13)(3,18,19,17)(4,21)(8,26,22,25)(9,15,27,28)(10,31)(11,32)(12,14)(16,20)(23,30,24,29);(1,30,21,23)(2,17,31,8)(3,12,25,10)(4,24,6,29)(5,27,20,15)(7,9,16,28)(11,19,13,26)(14,18,32,22)

Polycyclic presentation:

3,2^10|b^2=f,c^2=fgh,d^2=ghi,e^2=hi,b^a=bc,c^a=bhj,c^b=cfg,d^a=eghjk,d^b=dfghi,d^c=dfijk,e^a=deijk,e^b=efhijk,e^c=ehjk,e^d=eijk,f^a=fhk,f^c=fj,f^d=fjk,f^e=fk,g^a=ghjk,g^b=gj,g^c=gjk,h^a=gk,h^b=hjk,h^c=hk,i^a=ij,i^b=ij,i^c=ik,j^a=jk,k^a=j

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226653
maximal normal subgroup of#72226653
maximal quotient of#72267785, #72267828, #72267869

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