Finite Group Databasebeta

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#72341132

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237777
commutator#1781C22×C4
Frattini, cosocle#20585C23
prime radical#72285341C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1899 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1899 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=ghk,c^2=hik,e^2=k,f^2=gk,b^a=bdfgk,c^a=ce,c^b=cghik,d^b=dg,d^c=dij,e^a=eijk,e^b=egh,e^c=ei,f^a=fk,f^b=fik,f^c=fghi,f^e=fik,g^b=gk,g^c=gjk,g^e=gk,h^a=hi,h^b=hk,h^c=hk,h^d=hk,h^e=hk,h^f=hj,i^a=ik,i^b=ij,i^c=ik,i^f=ik,j^a=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341156
maximal normal subgroup of#72341156

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