Finite Group Databasebeta

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#72338959

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72262838
commutator#2297C22×C4
Frattini, cosocle#35527C23
prime radical#417807C22
LCSG ▷ #2297 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35527 ◁ G
LECSG ▷ #35527 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1

2381 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 2381 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^11|a^2=hj,b^2=f,c^2=dfhi,d^2=k,e^2=j,g^2=k,b^a=be,c^a=ce,c^b=cd,d^a=dg,d^b=dij,d^c=di,e^b=eh,e^c=eghk,e^d=ei,f^a=fhj,f^c=fijk,f^d=fk,f^e=fj,g^b=gi,g^c=gi,g^d=gk,g^f=gk,h^b=hj,h^c=hjk,h^d=hk,i^b=ik,j^c=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338976
maximal normal subgroup of#72338976

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