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#72271787

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#72268210C2
Fitting2−1+42D5
LCSG ▷ #72268210
UCSC1 ◁ C22

At least 24645 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has at least 24645 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) = 96
transitive degreeμt(G) = 1280

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

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

Polycyclic presentation:

2,5,2^10|a^2=k,e^2=k,f^2=k,g^2=l,h^2=k,i^2=k,j^2=kl,b^a=b^4,c^a=cfk,c^b=d,d^a=cefk,d^b=cde,d^c=dk,e^a=cdk,e^b=f,f^a=fk,f^b=cd,f^c=fk,f^e=fk,g^a=cdghk,g^b=ghij,g^c=gk,g^d=gk,g^e=gk,g^f=gk,h^a=cdehk,h^b=i,h^d=hk,h^e=hk,h^f=hk,h^g=hk,i^a=dgj,i^b=gil,i^c=ik,i^f=ik,i^g=il,i^h=il,j^a=defghil,j^b=gijkl,j^d=jk,j^h=jkl,j^i=jk

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272136

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