Finite Group Databasebeta

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#72323239

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216503
commutator, Frattini, cosocleC2×#2613C23
LCSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2613 ◁ G
LECSG ▷ C2×#2613 ▷ C25 ▷ C23 ▷ C2 ▷ C1

4762 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 4762 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=fhijk,b^a=bdik,c^a=ce,c^b=cef,d^c=dg,e^a=eg,e^b=eh,e^c=ehjk,e^d=ek,f^a=fhijk,f^b=fhj,f^c=fjk,f^d=fi,f^e=fj,g^a=gk,g^b=gi,g^c=gi,g^f=gk,h^a=hik,h^c=hj,i^c=ik,j^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72323241, #72324321, #72324333
maximal normal subgroup of#72323241, #72324321, #72324333

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