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#72267871

order5120
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2#72226628
commutator, prime radical, cosocle, hypocenter#72267830C2
Frattini, socleC25(C24⋊C5)⋊C2
Fitting#160295D5
LCSG ▷ #72267830
UCSC1 ◁ C2

786 conjugacy classes of subgroups; only the 7 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 7 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) = 80
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 2 generators on 80 points:

(1,32)(2,20,12,23)(3,13,19,35,17,36,46,48)(4,24,21,22)(5,9)(6,11)(7,10)(8,52,29,65)(14,77,41,68,42,73,39,80)(15,43,47,44,18,34,16,66)(25,33,53,30)(26,54,60,72)(27,61,31,79)(28,59,50,51)(37,75,69,40,55,45,67,49)(38,64,76,58)(56,70,63,74)(57,78,62,71);(1,17,14,38,53)(2,8,26,37,13)(3,42,54,25,7)(4,46,49,78,52)(5,18,39,71,61)(6,15,41,70,79)(9,30,63,73,35)(10,31,64,68,44)(11,33,57,77,48)(12,29,58,55,36)(16,75,72,51,23)(19,40,74,65,21)(20,47,45,76,59)(22,50,56,69,43)(24,28,62,67,34)(27,60,80,66,32)

Polycyclic presentation:

2,5,2^9|a^2=ij,c^2=g,d^2=i,e^2=gijk,f^2=g,b^a=b^4i,c^a=cij,c^b=efg,d^a=cfhjk,d^b=egjk,d^c=di,e^a=ceijk,e^b=ceij,e^c=eij,e^d=ejk,f^a=cdhjk,f^b=degijk,f^c=fgh,f^d=fghij,f^e=fgijk,h^a=gi,h^b=ghj,i^a=gh,i^b=gijk,j^a=ghij,j^b=ghij,k^a=gijk,k^b=gh

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267872

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