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#72268121

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268106 ⋊ C3
derived#8930496 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#35346⋊C3
commutator, cosocle, hypocenter#8930496C2×C3
FrattiniC25C2×C22⋊A4
Fitting#72268106C3
hypercenterC23#85645
LCSG ▷ #8930496
UCSC1 ◁ C22 ◁ C23

2017 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 2017 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) = 48
transitive degreeμt(G) = 128

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

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

Polycyclic presentation:

2,3,2^9|c^2=i,d^2=i,e^2=jk,f^2=jk,c^a=ch,c^b=cd,d^a=dg,d^b=c,d^c=di,e^b=ef,e^c=ek,e^d=eij,f^b=e,f^c=fijk,f^d=fk,f^e=fjk,g^b=h,g^c=gjk,g^d=gk,g^e=gk,h^b=ghj,h^c=hk,h^d=hj,h^f=hk,i^a=ik

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72268124

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