Finite Group Databasebeta

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#72241458

order10240
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

FittingH(5,𝔽4) ⋊ D5
derived#72241456 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22#72209701
commutator, cosocle, hypocenter#72241456C2
FittingH(5,𝔽4)D5
LCSG ▷ #72241456
UCSC1 ◁ C22

53095 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 53095 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

transitive degreeμt(G) = 64

Polycyclic presentation:

2,5,2^10|g^2=l,h^2=kl,i^2=k,j^2=k,b^a=b^4,c^a=def,c^b=cf,d^b=cd,e^a=de,e^b=de,f^a=ce,f^b=e,g^a=ceghj,g^b=gj,g^c=gl,g^d=gl,g^e=gk,h^a=defijl,h^b=ghl,h^d=hkl,h^e=hkl,h^f=hl,h^g=hl,i^a=di,i^b=hikl,i^c=ikl,i^e=ik,i^f=il,i^g=il,i^h=ikl,j^a=cdhikl,j^b=i,j^c=jk,j^d=jkl,j^e=jk,j^f=jk,j^g=jk,j^h=jk,j^i=jkl

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72242443, #72244262

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