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#72286121

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237427
commutatorC22×H(5,𝔽2)C22×C4
Frattini, cosocle#20586C23
prime radical#72285917C2
LCSG ▷ C22×H(5,𝔽2) ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6432959 ◁ G
LECSG ▷ #20586 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

3248 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 3248 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,9)(4,18)(5,11)(6,28)(7,13)(8,14)(10,21)(12,31)(15,39)(16,33)(17,35)(19,53)(20,44)(22,51)(23,59)(24,52)(25,41)(26,61)(27,34)(29,43)(30,63)(32,46)(36,56)(37,57)(38,55)(40,58)(42,49)(45,47)(48,50)(54,64)(60,62);(1,3)(2,9)(4,19)(5,25)(6,29,31,16)(7,27)(8,17)(10,36)(11,41)(12,43,28,33)(13,34)(14,35)(15,22,47,30)(18,50)(20,49,57,26)(21,54)(23,60,46,55)(24,48)(32,38,59,62)(37,42,44,61)(39,51,45,63)(40,64)(52,53)(56,58);(1,4,20,56)(2,10,37,53,14,40,44,50)(3,15,26,55)(5,21,29,19,27,58,16,48)(6,30,25,46,31,22,7,23)(8,24,57,54)(9,32,42,63,35,59,61,51)(11,18,33,36)(12,38,13,45)(17,47,49,60)(28,62,41,39)(34,52,43,64)

Polycyclic presentation:

2^11|b^2=g,c^2=efhk,f^2=hk,b^a=bgjk,c^a=cd,c^b=ce,d^b=dj,d^c=dg,e^a=eij,e^b=ei,e^c=egh,f^a=fgik,f^b=fghijk,f^c=fgh,f^d=fik,f^e=fij,g^c=gi,g^e=gk,g^f=gjk,h^a=hj,h^b=hj,h^c=hjk,h^d=hk,h^e=hk,i^b=ik,i^c=ij,i^f=ik,j^c=jk

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72286183, #72286190
maximal normal subgroup of#72286183, #72286190

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