Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C24×C27⋊C3 ⋊ C2 |
| Sylow | C27⋊C3 ⋊ #120 |
654 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 654 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 35 points:
(6,8)(9,34,28,32,17,31,33,16,26,12,25,27,11,14,13,18,30,23,21,29,35,20,10,22,19,24,15);(2,4);(1,2)(3,4)(5,6)(7,8)(10,13)(11,18)(12,21)(14,22)(15,25)(16,28)(17,31)(19,32)(20,33)(23,29)(24,27)(26,34)(30,35)Polycyclic presentation:
2^3,3,2^2,3^3|g^3=h,h^3=i,b^a=be,c^a=cf,g^a=g^2h^2i^2,g^d=gi^2,h^a=h^2i^2,i^a=i^2282 conjugacy classes — view the full character table.
| maximal subgroup of | #120×D27⋊C3 |
| maximal normal subgroup of | #120×D27⋊C3 |
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, 2, 3, 2, 2, 3, 3, 3]);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);