Finite Group Databasebeta

← Prev (#72228060) Next (#72228062) →

#72228061

order3072
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11715850
commutator, prime radical, cosocle, hypocenter#72227582C2
Frattini, hypercenterC23#87588
Fitting#8934620D3
LCSG ▷ #72227582
UCSC1 ◁ C22 ◁ C23

2326 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 2326 conjugacy classes of subgroups — too many to draw. Only its 28 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:

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

Polycyclic presentation:

2,3,2^9|a^2=ik,c^2=i,d^2=i,e^2=ik,f^2=ik,g^2=k,h^2=k,b^a=b^2,c^a=cd,c^b=d,d^a=di,d^b=cdi,d^c=di,e^a=ef,e^b=f,e^d=ek,f^a=fik,f^b=efik,f^c=fk,f^e=fik,g^a=gh,g^b=h,g^c=gk,g^d=gij,g^e=gk,g^f=gij,h^a=hk,h^b=ghk,h^c=hijk,h^d=hk,h^e=hijk,h^f=hk,h^g=hk,j^a=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228081
maximal normal subgroup of#72228081

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