Finite Group Databasebeta

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#72227046

order4096
nilpotency class6
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227045
commutatorC162C42
Frattini, cosocleC16×M64C22
prime radicalC23#93868
LCSG ▷ C162 ▷ 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

438 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 129 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^12|a^2=d,b^2=e,c^2=fghjkl,d^2=gjl,f^2=hj,g^2=ikl,h^2=jl,i^2=kl,j^2=l,k^2=l,b^a=bc,c^a=cf,c^b=cfgil,d^b=dgijk,e^a=ejkl,e^d=el,f^a=fh,f^b=fhl,g^a=gl,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jl,j^b=jl,k^b=kl

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227059, #72227127, #72227136
maximal normal subgroup of#72227059, #72227127, #72227136

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