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#72341209

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#251883
commutator, Frattini, cosocleC23×D4C24
prime radical#7266801C2
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

1510 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 1510 conjugacy classes of subgroups — too many to draw. Only its 127 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=j,c^2=fg,d^2=ghj,e^2=j,b^a=be,c^a=cf,c^b=cehj,d^a=dfhj,d^b=dgi,d^c=dfi,e^a=ej,e^b=ej,e^c=eh,e^d=ehi,f^b=fhj,f^c=fi,f^d=fij,g^a=gi,g^b=gj,g^c=gi,g^d=gj,g^e=gj,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341219, #72341236
maximal normal subgroup of#72341219, #72341236

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