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#72235366

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#8101
commutatorC25C2×C42
Frattini, cosocleC22×#124C23
prime radicalC22×H(3,𝔽4)C22
LCSG ▷ C25 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×H(3,𝔽4) ◁ G
LECSG ▷ C22×#124 ▷ C24 ▷ C22 ▷ C1

2016 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 2016 conjugacy classes of subgroups — too many to draw. Only its 143 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=dei,d^2=j,e^2=i,b^a=bf,c^a=cgh,c^b=cgj,d^b=dg,d^c=dj,e^a=eh,e^b=eg,e^c=ei,e^d=eij,f^a=fg,f^b=fh,f^d=fi,f^e=fij,g^a=gi,g^b=gij,h^a=hij,h^b=hj

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235372, #72235380
maximal normal subgroup of#72235372, #72235380

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