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#72341239

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237744
commutator, Frattini, cosocle#1781C24
prime radical#72310964C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2177 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 2177 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=k,b^2=ehijk,c^2=gk,d^2=hik,e^2=k,f^2=hjk,b^a=be,c^a=cgj,c^b=cf,d^a=dgij,d^b=dghijk,d^c=dgik,e^a=ek,e^b=ejk,e^c=eg,e^d=eijk,f^a=fgj,f^b=fgh,f^c=fhij,f^d=fijk,f^e=fijk,g^b=gi,g^d=gj,g^f=gjk,h^a=hjk,h^b=hj,h^c=hk,h^d=hk,h^e=hk,i^c=ij,i^d=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

47 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, 1, F.11);;
gap> SetPower(coll, 2, F.5*F.8*F.9*F.10*F.11);;
gap> SetPower(coll, 3, F.7*F.11);;
gap> SetPower(coll, 4, F.8*F.9*F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.8*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
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, 9, 3, F.9*F.10);;
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);