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#72268093

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268077 ⋊ C3
derived#8926711 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#35346⋊C3
commutator, cosocle, hypocenter#8926711C2×C3
FrattiniC25C2×C22⋊A4
Fitting#72268077C3
hypercenterC23#85645
LCSG ▷ #8926711
UCSC1 ◁ C22 ◁ C23

2028 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 2028 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)(3,7,11,6,4,15)(5,8)(9,16,10,13,12,14)(18,27,28,23,19,24)(20,21,34)(25,36,47,31,30,48)(26,46,45,35,41,38)(29,39,43,44,42,40)(32,33,37);(1,10,5,11)(2,14,8,15)(3,12,9,4)(6,16,13,7)(17,32,22,34)(18,40,23,43)(19,46,27,41)(20,37,33,21)(24,47,28,48)(25,35,31,26)(29,36,44,30)(38,42,45,39);(3,9)(7,16)(10,11)(14,15)(17,24,22,28)(18,27,23,19)(20,45,33,38)(21,39,37,42)(25,29,31,44)(26,36,35,30)(32,48,34,47)(40,46,43,41)

Polycyclic presentation:

2,3,2^9|c^2=j,d^2=j,e^2=jk,f^2=jk,c^a=ch,c^b=cd,d^a=dg,d^b=c,d^c=dj,e^b=ef,e^c=ejk,e^d=ei,f^b=e,f^c=fijk,f^d=fjk,f^e=fjk,g^b=h,g^c=gj,g^e=gk,h^b=ghj,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#72268096

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.10);;
gap> SetPower(coll, 4, F.10);;
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.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
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.10*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);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.10);;
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);