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#72316111

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227768
commutatorC2×#518C22×C4
Frattini, cosocle#35348C23
prime radical#6500015C22
LCSG ▷ C2×#518 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #35348 ◁ G
LECSG ▷ #35348 ▷ C25 ▷ C23 ▷ C2 ▷ C1

2497 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 2497 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^11|b^2=j,c^2=f,e^2=j,b^a=bd,c^a=cgi,c^b=ce,d^b=dk,d^c=dg,e^a=eg,e^b=ej,e^c=eh,e^d=ek,f^a=fik,f^b=fhj,f^d=fi,f^e=fj,g^c=gi,g^e=gk,g^f=gk,h^a=hik,h^c=hj,h^d=hk,i^b=ik,i^c=ik,j^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72316147
maximal normal subgroup of#72316147

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