Finite Group Databasebeta

← Prev (#72227519) Next (#72227521) →

#72227520

order5120
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

derivedC24⋊(C24⋊C5) ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC24⋊(C24⋊C5)C4
FrattiniC25(C24⋊C5)⋊C2
Fitting#6342371D5
socleC24#66245
prime radical, cosocle#72220361C2
LCSG ▷ C24⋊(C24⋊C5)
UCSC1

21666 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 21666 conjugacy classes of subgroups — too many to draw. Only its 8 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:

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

Polycyclic presentation:

2^2,5,2^8|a^2=b,c^a=c^4,d^a=defk,d^b=dhi,d^c=de,e^a=gjk,e^b=ehk,e^c=fg,f^a=efghi,f^b=fhj,f^c=df,g^a=ehj,g^b=gi,g^c=dfg,h^a=ij,h^c=ijk,i^a=hk,i^c=j,j^a=k,j^c=k,k^a=j,k^c=hk

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227521, C2×#72227520
maximal normal subgroup ofC2×#72227520
maximal quotient of#72239935, C2×#72227520, #72268897

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