Finite Group Databasebeta

← Prev (#72340951) Next (#72340953) →

#72340952

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72340950 ⋊ C3
derived#346283 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#346283C2×C3
FrattiniC25C2×C22⋊A4
Fitting#72340950C3
socleC22C2×#73293
LCSG ▷ #346283
UCSC1

1415 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 1415 conjugacy classes of subgroups — too many to draw. Only its 85 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:

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

Polycyclic presentation:

2,3,2^9|c^2=h,d^2=hj,e^2=ghik,f^2=ghij,c^a=cjk,c^b=cd,d^a=dj,d^b=cjk,d^c=dhk,e^a=ej,e^b=ef,e^c=ei,e^d=eg,f^a=fk,f^b=ek,f^c=fgik,f^d=fik,f^e=fghijk,g^b=gjk,g^c=gj,g^d=gk,g^e=gj,g^f=gk,h^b=hjk,h^e=hk,h^f=hjk,i^b=ij,i^c=ik,i^d=ijk,i^e=ijk,i^f=ij,j^b=jk,k^b=j

Conjugacy classes & character table

61 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, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.11);;
gap> SetPower(coll, 6, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.3*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);