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#72340997

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237593
commutator#2308C22×C4
Frattini, cosocle#20585C23
prime radical#6432953C22
LCSG ▷ #2308 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6432953 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3409 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 3409 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=gjk,f^2=gik,b^a=bd,c^a=cefgik,c^b=cg,d^a=dijk,d^c=dg,e^b=eijk,e^c=egh,f^a=fhj,f^b=fgik,f^c=fhjk,f^e=fik,g^a=gjk,g^e=gk,g^f=gk,h^a=hi,h^b=hjk,h^c=hk,h^d=hk,h^e=hk,h^f=hj,i^a=ik,i^c=ij,i^f=ik,j^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341003
maximal normal subgroup of#72341003

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