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#72227082

order4096
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227079
commutator, Frattini, cosocleC16×C32C23
prime radical#72225590C2
LCSG ▷ C16×C32 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ C16×C32 ◁ G
LECSG ▷ C16×C32 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

669 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,9)(4,10)(5,12)(6,14)(7,8)(11,43)(13,22)(15,23)(16,26)(17,30)(18,32)(19,34)(20,36)(21,28)(24,40)(25,29)(27,31)(33,60)(35,50)(37,49)(38,55)(39,63)(41,59)(42,58)(44,52)(45,57)(46,64)(47,51)(48,53)(54,62)(56,61);(2,8)(3,17)(4,22)(5,23)(6,26)(9,11)(10,32)(12,33)(13,29)(15,31)(18,25)(19,54)(20,57)(21,43)(24,53)(27,60)(28,30)(34,46)(35,39)(36,47)(37,44)(38,45)(40,41)(42,48)(49,50)(51,55)(52,63)(56,64)(58,59)(61,62);(2,9,8,28)(3,17)(4,22)(5,23)(6,26)(10,35,29,44)(11,13,30,32)(12,37,31,39)(14,38,16,36)(15,34,33,61)(18,25)(19,54)(20,57)(21,43)(24,53)(27,60)(40,46,42,62)(41,45,48,47)(49,50)(51,55)(52,63)(56,64)(58,59)

Polycyclic presentation:

2^12|c^2=j,d^2=fh,e^2=gil,f^2=hj,g^2=ik,h^2=j,i^2=kl,k^2=l,b^a=bl,c^a=ce,c^b=cd,d^a=dg,d^b=df,d^c=df,e^a=egl,e^b=eg,e^c=eg,f^a=fi,f^b=fh,f^c=fh,g^a=gi,g^b=gi,g^c=gi,h^a=hk,h^b=hj,h^c=hj,i^a=ik,i^b=ik,i^c=ik,j^a=jl,k^a=kl,k^b=kl,k^c=kl

Conjugacy classes & character table

310 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227093
maximal normal subgroup of#72227093

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 5, F.7*F.9*F.12);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11*F.12);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.12);;
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.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 2, F.11*F.12);;
gap> SetConjugate(coll, 11, 3, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);