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#72266654

order6144
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72266593 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4#72209707
commutator, Fitting, cosocle, hypocenter#72266593C3
FrattiniC25×C4C22⋊A4
socleC23#16297⋊C3
LCSG ▷ #72266593
UCSC1 ◁ C4

3038 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 3038 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 96

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

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

Polycyclic presentation:

3,2^11|b^2=f,c^2=fg,d^2=fgi,e^2=fkl,f^2=l,b^a=bc,c^a=bgik,c^b=cfi,d^a=de,d^b=dgi,d^c=dgh,e^a=dikl,e^b=eghjk,e^c=egk,e^d=efijkl,f^a=fgijkl,f^c=fk,f^d=fj,g^a=gikl,g^b=gjkl,g^c=gk,g^d=gjk,g^e=gk,h^a=ghl,h^b=hjkl,h^d=hk,h^e=hk,i^a=gj,i^b=ikl,i^c=ij,i^d=ik,i^e=ijl,j^a=jkl,k^a=j

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72266657

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