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#72267872

order10240
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC22#72226628
commutator, prime radical, cosocle, hypocenter#72267832C2
Frattini, socleC26(C24⋊C5)⋊C2
Fitting#72267796D5
LCSG ▷ #72267832
UCSC1 ◁ C22

3615 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 3615 conjugacy classes of subgroups — too many to draw. Only its 13 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) = 104
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 2 generators on 104 points:

(1,20)(2,4)(3,21,12,22)(5,7)(6,15,19,35)(8,14)(9,11,23,13)(10,39,28,37)(16,40,34,25,33,38,36,31)(17,30,18,29)(24,26)(27,32)(41,99,92,101)(42,82,90,55)(43,79,102,94)(44,97,72,77)(45,76,67,81)(46,50,58,80)(47,86,95,70)(48,53,98,96)(49,87,91,54)(51,64,73,59)(52,84,103,88)(56,65,104,89)(57,62,75,83)(60,100,63,69)(61,74,68,78)(66,93,85,71);(1,12,28,25,15)(2,6,33,32,9)(3,10,31,30,5)(4,11,27,38,29)(7,19,34,37,22)(8,17,16,24,23)(13,26,40,35,14)(18,36,39,21,20)(43,45,61,95,85)(44,46,66,72,54)(48,50,76,103,94)(49,51,79,91,69)(53,63,62,60,68)(55,84,59,83,102)(56,64,58,92,96)(57,77,75,81,65)(67,87,104,88,82)(70,99,74,97,98)(71,78,73,90,89)(80,100,93,101,86)

Polycyclic presentation:

2,5,2^10|a^2=h,c^2=g,d^2=j,e^2=ghl,f^2=g,b^a=b^4jl,c^a=cg,c^b=cd,d^a=cfil,d^b=cehl,d^c=dj,e^a=ceh,e^b=cfgl,e^c=eh,e^d=ehjl,f^a=cdhjl,f^b=cijk,f^c=fgi,f^d=fgijk,f^e=fghl,h^b=k,i^a=gj,i^b=gj,j^a=gi,j^b=l,k^a=gijk,k^b=hjl,l^a=ghl,l^b=ik

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267881

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", "l");;
gap> coll := SingleCollector(F, [2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.12);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.10*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.3*F.6*F.9*F.12);;
gap> SetConjugate(coll, 4, 2, F.3*F.5*F.8*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.3*F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.3*F.6*F.7*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10*F.12);;
gap> SetConjugate(coll, 6, 1, F.3*F.4*F.8*F.10*F.12);;
gap> SetConjugate(coll, 6, 2, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.12);;
gap> SetConjugate(coll, 8, 2, F.11);;
gap> SetConjugate(coll, 9, 1, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.7*F.10);;
gap> SetConjugate(coll, 10, 1, F.7*F.9);;
gap> SetConjugate(coll, 10, 2, F.12);;
gap> SetConjugate(coll, 11, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.8*F.10*F.12);;
gap> SetConjugate(coll, 12, 1, F.7*F.8*F.12);;
gap> SetConjugate(coll, 12, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);