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#72267848

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#125654
commutator, Frattini, cosocle#7554C23
prime radical#72267846C2
LCSG ▷ #7554 ▷ C24×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C24×C4 ◁ #7554 ◁ G
LECSG ▷ #7554 ▷ C24×C4 ▷ C23 ▷ C2 ▷ C1

3204 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 3204 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 48
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6)(7,14)(8,12)(9,15)(10,16)(17,18,22,26)(19,27,31,23)(20,33,35,24)(21,37,28,25)(29,46,48,42)(30,40,32,41)(34,45,47,39)(36,43,38,44);(1,2)(3,9,11,7)(4,12,13,6)(5,8)(10,15)(14,16)(17,19,28,32,22,31,21,30)(18,23,37,41,26,27,25,40)(20,34,38,48,35,47,36,29)(24,42,44,45,33,46,43,39);(1,3)(2,7)(4,10,13,16)(5,11)(6,14,12,15)(8,9)(17,20,21,38,22,35,28,36)(18,24,25,44,26,33,37,43)(19,29,32,47,31,48,30,34)(23,39,41,46,27,45,40,42)

Polycyclic presentation:

2^11|a^2=j,b^2=fgjk,c^2=fhk,d^2=g,e^2=h,f^2=j,b^a=be,c^a=cd,c^b=cfghijk,d^a=dg,d^b=df,d^c=dgi,e^a=eh,e^b=ehi,e^c=efjk,e^d=eijk,f^a=fi,f^b=fj,f^c=fjk,f^e=fk,g^b=gj,h^c=hjk,i^b=ik

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267867

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