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#72227762

order5120
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC2#72209701
commutator, prime radical, cosocle, hypocenter#72210028C2
FittingH(9,𝔽2)D5
LCSG ▷ #72210028
UCSC1 ◁ C2

At least 6610 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has at least 6610 conjugacy classes of subgroups — too many to draw. Only its 10 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2,5,2^9|a^2=k,g^2=k,i^2=k,b^a=b^4,c^a=cf,c^b=d,d^a=cef,d^b=cde,e^a=cd,e^b=f,f^b=cd,g^a=ghik,g^b=gh,g^c=gk,g^d=gk,g^e=gk,h^a=j,h^b=ij,h^f=hk,i^a=hijk,i^b=gik,i^d=ik,i^e=ik,i^f=ik,i^g=ik,j^a=h,j^b=gij,j^c=jk,j^d=jk,j^f=jk,j^h=jk,j^i=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227856, #72265383
maximal normal subgroup of#72265383
maximal quotient of#72268150, #72271305, #72271628, …

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