Finite Group Databasebeta

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#72271582

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

Aliases

No aliases for this group.

Splittings

Fitting2−1+42 ⋊ D5
derived#72268210 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22#72209701
commutator, cosocle, hypocenter#72268210C2
Fitting2−1+42D5
LCSG ▷ #72268210
UCSC1 ◁ C22

At least 12932 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has at least 12932 conjugacy classes of subgroups — too many to draw. Only its 15 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) = 64
transitive degreeμt(G) = 1024

A minimal faithful permutation representation — 2 generators on 64 points:

(1,6,2,20,9,27,10,14)(3,23,19,21,22,30,12,24)(4,17,11,28,8,18,16,32)(5,25,13,29,26,15,31,7)(33,59,48,46,41,55,40,45)(34,63,39,38,50,56,49,62)(35,64,36,37,42,60,53,61)(43,51,58,54,52,47,44,57);(2,11,3,18,21)(4,6,5,14,23)(7,28,31,19,15)(8,27,26,20,30)(10,16,22,17,24)(12,25,29,32,13)(34,43,42,47,57)(35,51,54,50,52)(36,49,37,40,63)(38,46,44,55,64)(39,61,48,56,53)(45,58,59,60,62)

Polycyclic presentation:

2,5,2^10|c^2=k,d^2=l,e^2=kl,f^2=kl,g^2=l,h^2=kl,i^2=k,j^2=kl,b^a=b^4,c^a=cdfhijkl,c^b=cf,d^a=efhl,d^b=cdk,d^c=dk,e^a=eghkl,e^b=del,e^c=ek,e^d=el,f^a=dehjk,f^b=e,f^c=fkl,f^d=fkl,f^e=fl,g^a=hjl,g^b=gh,g^c=gl,g^d=gkl,g^e=gkl,g^f=gl,h^a=gjl,h^b=gjk,h^d=hl,h^e=hk,h^f=hl,h^g=hkl,i^a=ghik,i^b=hijkl,i^c=ik,i^d=il,i^e=il,i^f=il,i^g=ik,i^h=ikl,j^a=jkl,j^b=ijk,j^c=jkl,j^e=jk,j^f=jkl,j^g=jl,j^h=jl,j^i=jk

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271711

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