Finite Group Databasebeta

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#72284727

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237816
commutator#2308C22×C4
Frattini, cosocle#20585C23
prime radical#72284704C2
LCSG ▷ #2308 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6432954 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3493 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 3493 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,15)(4,10)(5,11)(6,27)(7,31)(8,14)(9,17)(12,30)(13,25)(16,34)(18,52)(19,54)(20,41)(21,57)(22,38)(23,39)(24,40)(26,44)(28,43)(29,45)(32,37)(33,47)(35,50)(36,51)(42,59)(46,55)(48,62)(49,60)(53,61)(56,63)(58,64);(1,3,6,18)(2,9,12,34)(4,21,22,58)(5,17,26,16)(7,32,29,53)(8,20,30,47)(10,37,38,63)(11,41,42,52)(13,35,46,62)(14,36,27,60)(15,44,33,39)(19,55,56,25)(23,51,59,49)(24,50,28,48)(31,57,45,64)(40,54,43,61);(1,4)(2,10)(3,16,20,49)(5,25)(6,28,30,22)(7,23)(8,24)(9,33)(11,13)(12,43,27,38)(14,40)(15,34,41,60)(17,18)(19,48)(21,56,50,53)(26,29,59,55)(31,39)(32,58)(35,61,57,63)(36,52)(37,62)(42,45,44,46)(47,51)(54,64)

Polycyclic presentation:

2^11|b^2=e,c^2=gj,d^2=h,f^2=hijk,b^a=bd,c^a=cgk,c^b=cefhijk,d^a=dh,d^b=dg,d^c=dik,e^a=eghjk,e^c=eghk,e^d=ei,f^a=fghk,f^b=fgk,f^c=fgij,f^d=fij,f^e=fi,g^b=gi,g^d=gjk,g^e=gk,g^f=gjk,h^b=hjk,h^c=hk,h^e=hk,h^f=hk,i^a=ij,i^b=ik,i^c=ijk,i^d=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

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