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#72284035

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#20585C23
prime radical#72284006C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2124 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2124 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:

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

Polycyclic presentation:

2^11|a^2=hjk,b^2=gh,c^2=efhj,d^2=hjk,f^2=hj,b^a=bhi,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dghij,e^a=egik,e^b=eij,e^c=eg,e^d=ei,f^a=fk,f^b=fgik,f^c=fg,f^d=fk,f^e=fik,g^a=gjk,g^b=gj,g^c=gi,g^d=gj,g^f=gjk,h^b=hj,h^c=hj,h^e=hk,i^a=ij,i^b=ik,i^c=ijk,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#72285484
maximal normal subgroup of#72285484

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