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#72284516

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#251828C22
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1989 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1989 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=gi,d^2=k,e^2=h,f^2=hi,b^a=be,c^a=cf,c^b=cg,d^b=dg,d^c=dgi,e^a=egh,e^b=eh,e^c=egjk,e^d=eijk,f^a=fgh,f^b=fgj,f^c=fhk,f^d=fijk,f^e=fik,g^a=gi,g^c=gj,g^e=gjk,g^f=gj,h^a=hjk,h^c=hjk,h^d=hk,h^f=hk,i^b=ij,i^c=ij,i^e=ik,i^f=ik,j^a=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285855
maximal normal subgroup of#72285855

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