Finite Group Databasebeta

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#72311005

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72216452
commutator#1781C22×C4
Frattini, cosocle#20586C23
prime radical#72285966C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20586 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2130 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2130 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)(4,10)(5,11)(6,12)(7,8)(9,16)(14,41)(15,47)(17,54)(18,44)(19,42)(20,36)(21,39)(22,29)(23,38)(24,30)(25,37)(26,40)(27,31)(28,53)(32,51)(33,50)(34,45)(35,55)(43,56)(46,63)(48,60)(49,59)(52,58)(57,61)(62,64);(2,8)(3,14)(9,32)(10,29)(11,30)(12,31)(13,41)(16,51)(17,46)(18,19)(20,49)(23,26)(28,34)(33,47)(35,60)(37,39)(42,44)(43,61)(45,53)(48,55)(52,64)(54,63)(56,57)(58,62);(1,3,15,48,7,18,50,64)(2,9,33,61,8,28,47,63)(4,19,26,52)(5,16,25,46)(6,17,49,45,27,56,20,51)(10,32,40,43)(11,13,39,62)(12,35,59,41,31,58,36,42)(14,23,55,22)(21,57,24,53)(29,34,38,54)(30,44,37,60)

Polycyclic presentation:

2^11|c^2=defk,d^2=h,b^a=bjk,c^a=ce,c^b=cd,d^a=dg,d^b=dh,d^c=dghik,e^b=egij,e^c=egijk,e^d=eijk,f^a=fij,f^b=fh,f^c=fhjk,f^d=fjk,f^e=fk,g^b=gj,g^c=gi,g^d=gj,g^f=gk,h^a=hj,h^c=hjk,h^e=hk,i^b=ij,i^c=ijk,i^d=ik,j^c=jk

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311013, #72340325, #72340462, …
maximal normal subgroup of#72311013, #72340325, #72340462, …

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