Finite Group Databasebeta

← Prev (#72235870) Next (#72235872) →

#72235871

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72235856 ⋊ C3
derived#93142 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#93142C2×C3
FrattiniC22×#126C2×A4
Fitting#72235856C3
socleC22C2×#1304⋊C3
LCSG ▷ #93142
UCSC1

721 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 17 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,10,15,8,31,6)(2,9,7,14,5,36)(3,43,60)(4,45,55,30,54,63)(11,49,32,19,17,12)(13,34,33)(16,25,27,24,50,23)(18,52,48)(20,53,59,37,44,61)(21,57,28)(22,39,38)(26,58,64,29,41,62)(40,47,56,46,42,51);(1,63,13,49,5,62,35,17)(2,39,10,44,14,52,7,40)(3,23,54,60,22,32,58,48)(4,20,42,19,30,37,61,24)(6,43,36,53,15,57,31,46)(8,64,33,25,9,55,34,50)(11,26,51,59,16,29,56,47)(12,41,38,18,27,45,28,21)

Polycyclic presentation:

2,3,2^9|a^2=j,c^2=f,d^2=efgj,f^2=jk,g^2=k,b^a=befhik,c^a=ci,c^b=defij,d^a=dk,d^b=cde,d^c=de,e^a=ek,e^b=eik,e^c=eh,e^d=ei,f^a=fjk,f^b=efghij,f^d=fh,f^e=fj,g^a=gj,g^b=fjk,g^c=ghi,g^d=ghi,g^e=gjk,g^f=gk,h^b=ijk,h^c=hj,h^d=hjk,i^b=hik,i^c=ijk,i^d=ik,j^b=k,k^b=jk

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235874

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