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#72227445

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2#72209725
commutator, prime radical, cosocle, hypocenter#72227444C2
Frattini, socleC25(C24⋊C5)⋊C2
Fitting#153709D5
LCSG ▷ #72227444
UCSC1 ◁ C2

1133 conjugacy classes of subgroups; only the 7 normal subgroups are shown.

normal subgroups only

This group has 1133 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:

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

Polycyclic presentation:

2,5,2^9|a^2=ghi,c^2=g,d^2=h,e^2=i,f^2=j,b^a=b^4ik,c^a=ei,c^b=dgh,d^a=dghi,d^b=ehi,e^a=cg,e^b=fgj,e^c=ek,f^a=cdefgk,f^b=cdefj,f^c=fk,f^d=fk,g^a=i,g^b=h,h^b=i,i^a=g,i^b=j,j^a=ghijk,j^b=ghijk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227457, C2×#72227445, #72239180, …
maximal normal subgroup ofC2×#72227445, #72239180, #72239361, …
maximal quotient ofC2×#72227445, #72268647

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