Finite Group Databasebeta

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#72338973

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72262850
commutatorC2×#535C22×C4
Frattini, cosocle#35527C23
prime radical#449492C22
LCSG ▷ C2×#535 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35527 ◁ G
LECSG ▷ #35527 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1

2283 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 2283 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=fj,c^2=dehij,d^2=ijk,e^2=j,g^2=k,b^a=bef,c^a=cd,c^b=cdef,d^a=dg,d^b=dh,d^c=dgi,e^b=ehik,e^c=egij,e^d=eik,f^a=fhik,f^c=fik,f^d=fjk,f^e=fj,g^a=gi,g^b=gik,g^c=gi,g^d=gk,g^f=gk,h^a=hj,h^b=hjk,h^c=hjk,h^d=hk,i^b=ik,i^c=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.5);;
gap> SetPower(coll, 2, F.6*F.10);;
gap> SetPower(coll, 3, F.4*F.5*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.9*F.10*F.11);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.11);;
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, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
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, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);