Finite Group Databasebeta

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#72227198

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227197
commutatorC2×C32C22×C8
Frattini, cosocle#7330C23
prime radical#513082C4
LCSG ▷ C2×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×#1311 ◁ #149495 ◁ G
LECSG ▷ #7330 ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

405 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=ik,d^2=f,e^2=gik,f^2=i,g^2=h,h^2=j,i^2=k,j^2=k,b^a=be,c^b=ci,d^b=dhijk,e^a=eg,e^b=eghijk,e^d=ej,f^b=fj,f^e=fk,g^a=gh,g^b=ghjk,g^d=gk,h^a=hj,h^b=hjk,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227204
maximal normal subgroup of#72227204

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