Finite Group Databasebeta

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#72268536

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

Aliases

No aliases for this group.

Splittings

FittingH(5,𝔽4) ⋊ D5
derived#72268535 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22#72209701
commutator, cosocle, hypocenter#72268535C2
FittingH(5,𝔽4)D5
LCSG ▷ #72268535
UCSC1 ◁ C22

At least 48730 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has at least 48730 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

minimal degreeμ(G) = 64
transitive degreeμt(G) = 320

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

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

Polycyclic presentation:

2,5,2^10|c^2=k,d^2=k,g^2=l,h^2=kl,i^2=k,j^2=k,b^a=b^4,c^a=cdfhjl,c^b=cf,d^a=efghi,d^b=cdk,d^c=dk,e^a=egjl,e^b=dek,e^c=ek,e^d=ek,f^a=degl,f^b=e,f^e=fk,g^a=hijk,g^b=gi,g^c=gk,g^d=gkl,g^e=gk,g^f=gk,h^a=ghjk,h^b=ghjl,h^c=hl,h^e=hkl,h^f=hk,h^g=hkl,i^a=gijl,i^b=hikl,i^c=il,i^d=ikl,i^f=ikl,i^g=ik,i^h=ikl,j^a=ghikl,j^b=i,j^c=jk,j^d=jk,j^e=jkl,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#72269016

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.11);;
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.3*F.4*F.6*F.8*F.10*F.12);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.5*F.6*F.7*F.8*F.9);;
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.10*F.12);;
gap> SetConjugate(coll, 5, 2, F.4*F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.4*F.5*F.7*F.12);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.10*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.11*F.12);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11*F.12);;
gap> SetConjugate(coll, 9, 1, F.7*F.9*F.10*F.12);;
gap> SetConjugate(coll, 9, 2, F.8*F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.12);;
gap> SetConjugate(coll, 9, 4, F.9*F.11*F.12);;
gap> SetConjugate(coll, 9, 6, F.9*F.11*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.7*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);;
gap> SetConjugate(coll, 10, 5, F.10*F.11*F.12);;
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);