Finite Group Databasebeta

← Prev (#72241426) Next (#72241428) →

#72241427

order3072
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C2×C43

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72228237 ⋊ C3

Subgroups

subgroupG/H
center, socle, hypercenterC22#11709219
commutator, Fitting, cosocle, hypocenter#72228237C3
FrattiniC26C22⋊A4
LCSG ▷ #72228237
UCSC1 ◁ C22

1134 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 22 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:

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

Polycyclic presentation:

3,2^10|b^2=h,c^2=hi,d^2=jk,e^2=jk,b^a=ch,c^a=bc,d^a=de,d^b=df,d^c=dg,e^a=d,e^b=efg,e^c=efk,e^d=ejk,f^a=fgj,f^b=fj,f^c=fjk,g^a=f,g^b=gjk,g^c=gk,h^a=hi,h^d=hj,h^e=hk,i^a=h,i^d=ijk,i^e=ij

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72241454, #72243519, #72249582, …

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