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#72227155

order8192
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227154
commutatorC16×C32C22×C4
Frattini, cosocleC16×M64C23
prime radical#72216368C22
LCSG ▷ C16×C32 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ C16×M64 ◁ G
LECSG ▷ C16×M64 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

1422 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1422 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^13|a^2=f,c^2=fikl,d^2=gi,e^2=hj,f^2=m,g^2=ik,h^2=jl,i^2=k,j^2=lm,l^2=m,b^a=bk,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dgk,e^a=ehkm,e^b=eh,e^c=ehl,f^b=fm,f^c=fkm,f^e=fm,g^a=gj,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hjm,i^a=il,i^b=ik,i^c=ik,j^a=jl,j^b=jl,j^c=jl,k^a=km,l^a=lm,l^b=lm,l^c=lm

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227170
maximal normal subgroup of#72227170

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.6*F.9*F.11*F.12);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.13);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetPower(coll, 8, F.10*F.12);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12*F.13);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.2*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.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.11*F.13);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.13);;
gap> SetConjugate(coll, 6, 3, F.6*F.11*F.13);;
gap> SetConjugate(coll, 6, 5, F.6*F.13);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
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.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
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, 10, 2, F.10*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 11, 1, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> SetConjugate(coll, 12, 3, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);