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#72271713

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

Aliases

No aliases for this group.

Subgroups

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

At least 12720 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has at least 12720 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) = 96
transitive degreeμt(G) = 640

A minimal faithful permutation representation — 2 generators on 96 points:

(1,13,23,12)(2,22,28,18)(3,21,7,30)(4,25,32,24)(5,27,19,9)(6,26,31,20)(8,11,16,10)(14,15,29,17)(33,78,64,73)(34,39,91,58)(35,52,60,51)(36,80,43,53)(37,83,65,95)(38,79,59,87)(40,48,41,86)(42,90,63,89)(44,61,82,68)(45,75,74,57)(46,96,47,54)(49,66,94,67)(50,56,93,72)(55,85,71,84)(62,92,70,88)(69,77,81,76);(2,9,19,23,22)(3,16,15,14,21)(4,20,5,17,26)(6,24,7,18,25)(10,27,31,29,13)(11,12,30,32,28)(35,54,36,59,70)(37,63,38,65,57)(39,56,62,77,55)(40,71,76,60,75)(41,58,42,64,66)(44,78,45,84,91)(46,86,47,69,82)(48,80,79,94,74)(49,87,93,92,73)(50,81,51,83,53)(61,72,68,96,67)(85,95,90,88,89)

Polycyclic presentation:

2,5,2^10|a^2=kl,c^2=k,d^2=kl,e^2=l,f^2=l,g^2=l,h^2=kl,i^2=k,j^2=k,b^a=b^4,c^a=cdfk,c^b=cf,d^a=efl,d^b=cdk,d^c=dk,e^a=el,e^b=dekl,e^c=ek,e^d=ekl,f^a=dek,f^b=e,f^c=fl,f^d=fl,f^e=fkl,g^a=ghjl,g^b=gj,h^a=ijk,h^b=ghl,h^g=hl,i^a=ik,i^b=hikl,i^g=il,i^h=ikl,j^a=hil,j^b=i,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#72272090

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