Finite Group Databasebeta

← Prev (#72263258) Next (#72263260) →

#72263259

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23×C42

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72262934 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4#11709219
commutator, Fitting, cosocle, hypocenter#72262934C3
FrattiniC24×C4C22⋊A4
socleC2#72210054
LCSG ▷ #72262934
UCSC1 ◁ C4

3915 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 3915 conjugacy classes of subgroups — too many to draw. Only its 31 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:

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

Polycyclic presentation:

3,2^10|b^2=f,c^2=fi,d^2=k,e^2=k,f^2=k,g^2=k,b^a=cfgk,c^a=bcg,c^b=cg,d^a=e,d^b=dh,d^c=dhij,e^a=dek,e^b=ehij,e^c=eijk,e^d=ek,f^a=fik,f^c=fk,f^d=fk,h^a=ijk,h^b=hk,i^a=fg,i^b=ik,i^c=ik,i^e=ik,j^a=fghijk,j^e=jk

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72263267, #72267211

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