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#72267831

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

Aliases

No aliases for this group.

Splittings

Fitting#160295 ⋊ D5
derived#72267830 ⋊ C2

Subgroups

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

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

normal subgroups only

This group has 911 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) = 40
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 40 points:

(1,3,6,15,4,13,5,12)(2,33,14,32,9,16,10,17)(7,31,18,25,19,28,26,11)(8,20,29,38,24,37,22,36)(21,34,30,35,23,39,27,40);(1,2,8,20,7)(3,11,29,34,32)(4,9,24,37,19)(5,14,30,40,26)(6,10,27,35,18)(12,28,21,38,16)(13,25,22,39,17)(15,31,23,36,33)

Polycyclic presentation:

2,5,2^9|c^2=g,d^2=h,e^2=g,f^2=g,b^a=b^4,c^a=cdfijk,c^b=e,d^a=efgi,d^b=cdj,d^c=dj,e^a=eg,e^b=cdfgijk,e^c=ejk,e^d=eh,f^a=dehj,f^b=chj,f^c=fi,f^d=fghij,f^e=fik,h^a=ik,h^b=ghj,i^a=gj,i^b=jk,j^a=gi,j^b=hjk,k^a=ghj,k^b=ik

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267833

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