Finite Group Databasebeta

← Prev (#72267827) Next (#72267829) →

#72267828

order6144
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72267824 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72226640
commutator, Fitting, cosocle, hypocenter#72267824C3
FrattiniC25×C4C22⋊A4
socleC23#11709313
LCSG ▷ #72267824
UCSC1 ◁ C2

3326 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 3326 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) = 48
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,7)(3,12,34)(4,8,25)(5,11,24)(6,10,31)(9,28,38)(13,30,21)(14,36,22)(15,35,44)(16,39,45)(17,26,48)(18,27,47)(19,41,46)(20,33,23)(29,43,40)(32,42,37);(3,15,14,16)(4,17,18,19)(8,26)(9,29)(10,33)(12,36)(21,42)(22,45)(23,31)(27,41)(28,43)(30,32)(34,44)(46,48);(1,13)(2,36,11,12)(3,18)(4,14)(5,38)(6,40)(7,47,24,25)(8,9,27,30)(10,35,33,39)(15,17)(16,19)(20,37)(21,34,28,22)(23,48,31,46)(26,32,41,29)(42,44,43,45)

Polycyclic presentation:

3,2^11|b^2=jkl,c^2=jl,d^2=gj,e^2=f,h^2=l,b^a=cjkl,c^a=bc,d^a=egij,d^b=dg,d^c=dghl,e^a=dei,e^b=eghil,e^c=ef,e^d=ei,f^a=fgij,f^b=fj,g^a=fkl,g^c=gjk,h^a=ghikl,h^b=hjk,h^c=hjkl,h^d=hjkl,h^e=hjl,i^b=ijkl,i^c=ijl,j^a=kl,k^a=jkl

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267829
maximal normal subgroup of#72267829

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", "l");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10*F.11*F.12);;
gap> SetPower(coll, 3, F.10*F.12);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 8, F.12);;
gap> SetConjugate(coll, 2, 1, F.3*F.10*F.11*F.12);;
gap> SetConjugate(coll, 3, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.12);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.11*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.9*F.11*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 5, F.8*F.10*F.12);;
gap> SetConjugate(coll, 9, 2, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 3, F.9*F.10*F.12);;
gap> SetConjugate(coll, 10, 1, F.11*F.12);;
gap> SetConjugate(coll, 11, 1, F.10*F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);