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#72284129

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

1986 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1986 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)(4,18)(8,26)(10,32)(14,35)(15,37)(16,38)(17,19)(21,36)(27,54)(28,55)(33,40)(34,39)(41,59)(42,60)(43,47)(44,62)(45,48)(53,64)(61,63);(1,2)(3,14,17,33)(4,9,18,31)(5,10)(6,24,25,11)(7,12)(8,15,28,43)(13,35)(16,44)(19,40)(20,29,46,57)(21,50)(22,39,52,34)(23,32)(26,37)(27,42)(30,36)(38,59,48,62)(41,45)(47,55)(49,58)(51,56)(53,63)(54,61,64,60);(1,3,6,16)(2,8,11,27,12,28,24,53)(4,19,21,48)(5,15,22,42,23,43,52,63)(7,17,25,45)(9,26,30,54,31,55,50,64)(10,33,34,44)(13,36,38,18)(14,39,41,32)(20,37,49,60,46,47,51,61)(29,40,56,62)(35,58,59,57)

Polycyclic presentation:

2^11|b^2=ik,c^2=f,d^2=h,f^2=hk,b^a=bghjk,c^a=cd,c^b=ce,d^a=dh,d^b=dg,d^c=dh,e^a=eg,e^b=ej,e^c=egi,e^d=eij,f^b=fgi,f^e=fij,g^b=gk,g^c=gi,g^d=gjk,g^f=gj,h^b=hjk,h^e=hk,i^a=ij,i^c=ij,i^d=ik,i^f=ik,j^c=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, 2, F.9*F.11);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.8*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8*F.10*F.11);;
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.10);;
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, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*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, 9, 1, F.9*F.10);;
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);