Finite Group Databasebeta

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#72311026

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216452
commutator#1781C22×C4
Frattini, cosocle#20586C23
prime radical#251828C22
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2021 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2021 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)(3,13,18,40)(4,19,22,9)(5,10,23,34)(6,11,25,36)(7,12)(8,16)(14,26)(15,37,47,38)(17,39)(20,31)(21,32)(24,35)(27,48)(28,57,58,41)(29,49)(30,45)(33,51)(42,53)(43,54,44,55)(46,61)(50,62)(52,60)(56,64,59,63);(1,3,14,41)(2,8,26,39)(4,15,44,63)(5,16,6,17)(7,18,42,57)(9,30,54,61)(10,13,11,28)(12,29,53,62)(19,27,55,60)(20,45,21,46)(22,47,43,64)(23,49,25,50)(24,52,51,48)(31,38,32,59)(33,37,35,56)(34,40,36,58);(3,15)(8,27)(13,37)(14,42)(16,48)(17,46)(18,47)(20,51)(26,53)(28,56)(29,30)(31,33)(38,40)(39,61)(41,63)(45,49)(50,52)(57,64)(58,59)(60,62)

Polycyclic presentation:

2^11|a^2=ghik,b^2=ef,d^2=h,b^a=be,c^b=cd,d^a=dg,d^b=dgi,d^c=dh,e^a=eij,e^c=egik,e^d=eij,f^a=fij,f^c=fh,f^d=fjk,g^a=gjk,g^b=gi,g^c=gj,g^d=gj,g^f=gk,h^a=hj,h^b=hjk,h^e=hk,i^a=ik,i^b=ik,i^c=ij,i^d=ik,j^b=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311187, #72340325, #72340466
maximal normal subgroup of#72311187, #72340325, #72340466

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