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#72307653

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#9915
commutator, Frattini, cosocleC23×M16C23
prime radical#60444C22
LCSG ▷ C23×M16 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×M16 ◁ G
LECSG ▷ C23×M16 ▷ C22×C4 ▷ C2 ▷ C1

1415 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 1415 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 24
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,7)(4,6)(5,8)(10,15,19,17)(11,20)(12,13)(16,23,18,24)(21,22);(1,3)(2,5,6,7)(4,8)(9,10,11,17,13,18,21,24,14,19,22,15,12,16,20,23);(5,7)(10,16,19,18)(15,23,17,24)

Polycyclic presentation:

2^10|a^2=h,b^2=d,c^2=h,d^2=i,e^2=ij,g^2=j,i^2=j,b^a=be,c^a=cf,c^b=cdefhj,d^a=dg,d^c=dh,e^a=eij,e^b=egi,e^c=eh,e^d=ej,f^b=fghi,g^a=gj,g^b=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72307689
maximal normal subgroup of#72307689

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