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#72283944

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216452
commutator#1781C22×C4
Frattini, cosocle#20586C23
prime radical#72283904C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2072 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2072 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^11|b^2=ghj,c^2=efhijk,d^2=h,e^2=k,f^2=hijk,b^a=bghj,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dh,e^a=eg,e^b=ei,e^c=egi,e^d=eij,f^a=fg,f^b=fgj,f^c=fgi,f^d=fij,f^e=fijk,g^b=gjk,g^c=gi,g^d=gjk,g^f=gj,h^b=hjk,h^e=hk,h^f=hk,i^a=ij,i^b=ik,i^c=ij,i^d=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285547, #72340332, #72340395, …
maximal normal subgroup of#72285547, #72340332, #72340395, …

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