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#72267957

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C27×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72267950 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4C22⋊#5464
commutator, Fitting, cosocle, hypocenter#72267950C3
FrattiniC22×C4C22⋊(C22⋊A4)
socleC2#72210049
LCSG ▷ #72267950
UCSC1 ◁ C4

10494 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 10494 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

3,2^10|b^2=k,c^2=k,d^2=hk,e^2=hi,f^2=k,g^2=k,h^2=k,b^a=c,c^a=bck,c^b=ck,d^a=eij,d^b=dh,d^c=dhj,e^a=dehijk,e^b=ei,e^c=ehi,e^d=ehijk,f^a=g,f^c=fk,g^a=fgk,g^b=gk,g^f=gk,h^a=hi,h^b=hk,h^e=hk,i^a=ij,i^c=ik,i^d=ik,i^e=ik,j^a=i,j^b=jk,j^c=jk,j^d=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267958

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