Finite Group Databasebeta

← Prev (#72227272) Next (#72227274) →

#72227273

order4096
nilpotency class6
automorphism?
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227271
commutatorC162C2×C8
Frattini, cosocle#72216220C22
prime radical#72216236C4
LCSG ▷ C162 ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ #72216220 ◁ G
LECSG ▷ #72216220 ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

592 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 87 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:

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

Polycyclic presentation:

2^12|a^2=c,b^2=l,c^2=e,d^2=fghik,e^2=hjl,f^2=hl,g^2=ij,h^2=jl,i^2=k,j^2=l,k^2=l,b^a=bd,c^b=cf,d^a=dg,d^b=dfg,d^c=djk,e^b=eh,f^a=fjk,f^b=fhjl,f^c=fl,g^a=gi,g^b=gijk,g^c=gl,h^b=hj,i^a=ik,i^b=ikl,j^b=jl,k^a=kl,k^b=kl

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227285
maximal normal subgroup of#72227285

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