Finite Group Databasebeta

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#72268535

order5120
automorphism?
homologyAbelianization (H1) = C5 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting, derived, SylowH(5,𝔽4) ⋊ C5

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22C24⋊(C24⋊C5)
commutator, Fitting, cosocle, hypocenterH(5,𝔽4)C5
LCSG ▷ H(5,𝔽4)
UCSC1 ◁ C22

86159 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 86159 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 320

Polycyclic presentation:

5,2^10|b^2=j,c^2=j,d^2=j,e^2=j,f^2=k,g^2=j,h^2=j,i^2=jk,b^a=e,c^a=d,c^b=cj,d^a=bcd,d^c=dj,e^a=cdj,e^c=ej,e^d=ej,f^a=fg,f^b=fjk,f^c=fjk,f^d=fk,f^e=fjk,g^a=h,g^b=gk,g^c=gjk,g^d=gj,g^f=gj,h^a=fijk,h^b=hj,h^c=hjk,h^d=hjk,h^e=hk,h^f=hk,h^g=hjk,i^a=ghjk,i^b=ik,i^c=ik,i^d=ij,i^e=ik,i^g=ijk,i^h=ij

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72268536, #72271713
maximal subgroup of#72268536, #72271713
maximal normal subgroup of#72268536, #72271713

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