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#72339804

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72338281
commutator#1963C22×C4
Frattini, cosocle#21209C23
prime radical#72338246C2
LCSG ▷ #1963 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42 ◁ #21209 ◁ G
LECSG ▷ #21209 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

4588 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 4588 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,12)(4,9)(5,7)(6,10)(8,15)(11,20)(13,30)(14,23)(16,17)(18,21)(22,27)(24,26)(28,31);(3,13)(4,18)(6,20)(8,25)(9,27)(10,11)(12,29)(14,21)(15,28)(16,19)(17,30)(22,23)(24,32)(26,31);(1,3,14,31,7,17,9,28)(2,8,4,19)(5,24,23,29)(6,12,21,32,20,16,27,25)(10,26,22,13)(11,15,18,30)

Polycyclic presentation:

2^11|c^2=def,d^2=hk,e^2=k,f^2=k,g^2=j,h^2=k,i^2=k,b^a=bgk,c^a=ce,c^b=cd,d^a=dg,d^b=dh,d^c=dg,e^a=ek,e^b=egjk,e^c=egj,e^d=eijk,f^a=fj,f^b=fk,f^c=fj,f^e=fk,g^a=gj,g^b=gj,g^c=gi,g^d=gjk,g^e=gj,h^a=hk,h^b=hk,h^c=hk,i^a=ij,i^b=ijk,i^d=ik,i^e=ik,j^c=jk

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72339812
maximal normal subgroup of#72339812

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