Finite Group Databasebeta

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#72272594

order10240
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#72272519 ⋊ D5
derived#72272532 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22#72209701
commutator, cosocle, hypocenter#72272532C2
Fitting#72272519D5
LCSG ▷ #72272532
UCSC1 ◁ C22

48980 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 48980 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|c^2=l,d^2=k,e^2=kl,f^2=l,g^2=l,h^2=k,i^2=kl,j^2=kl,b^a=b^4,c^a=fl,c^b=ce,d^a=dk,d^b=cdl,d^c=dl,e^a=cdefk,e^b=defk,e^c=ekl,e^d=ekl,f^a=cl,f^b=c,f^c=fkl,f^d=fl,f^e=fl,g^a=efhk,g^b=efhkl,g^c=gk,g^d=gk,g^e=gk,h^a=defgk,h^b=ceik,h^e=hkl,h^f=hkl,h^g=hl,i^a=fhjk,i^b=cdefgijkl,i^d=il,i^f=il,i^h=ik,j^a=cdefgil,j^b=dgil,j^c=jk,j^e=jk,j^f=jkl,j^g=jk,j^h=jkl,j^i=jk

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272633
maximal normal subgroup of#72272633

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