Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
559 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 559 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,11,21,6,4,19,7,13,12,3,14,22)(2,16,24,5,9,23,10,15,17,8,18,20)(25,26,27)(28,31,36,30,34,32,33,29,35);(1,7)(2,10)(4,13)(9,16)(11,14)(12,23)(15,18)(17,19)(20,21)(22,24)(26,27)(29,31,34)(32,36,35)Polycyclic presentation:
2^2,3^3,2^2,3,2|a^2=i,b^2=i,d^3=e,f^2=i,g^2=i,b^a=bi,d^c=de,f^d=g,g^d=fgi,g^f=gi,h^a=h^2153 conjugacy classes — view the full character table.
| maximal subgroup of | #72253017 |
| maximal normal subgroup of | #72253017 |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 2, 2, 3, 2]);;
gap> SetPower(coll, 1, F.9);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);