Finite Group Databasebeta

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#72339063

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72262839
commutator#2297C22×C4
Frattini, cosocle#35527C23
prime radical#72339039C2
LCSG ▷ #2297 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35527 ◁ G
LECSG ▷ #35527 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1

2724 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 2724 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:

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

Polycyclic presentation:

2^11|a^2=hj,b^2=hj,c^2=f,e^2=j,g^2=k,c^a=ce,c^b=cd,d^a=dg,d^b=dj,d^c=dijk,e^b=egi,e^c=eh,e^d=ei,f^a=fhj,f^b=fijk,f^d=fk,f^e=fj,g^c=gi,g^d=gk,g^f=gk,h^b=hk,h^c=hj,h^d=hk,i^b=ik,i^c=ik,j^b=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72339071
maximal normal subgroup of#72339071

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