Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C3 ⋊ #144388 |
2844 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 2844 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,3,4,7)(2,5)(6,8)(9,11,12,18)(10,14,15,23)(13,19,22,20)(16,17,21,24)(25,26,27);(5,8)(11,18)(14,23)(17,24)(19,20);(1,2,4,6)(3,5)(7,8)(9,10)(11,17,19,14)(12,21)(13,16)(15,22)(18,24,20,23)(26,27)Polycyclic presentation:
2^9,3|a^2=d,c^2=degi,b^a=bdhi,c^a=cf,c^b=cde,d^c=dg,e^a=eh,e^c=eg,f^a=fg,f^b=fgh,f^c=fh,f^e=fi,h^c=hi,j^a=j^290 conjugacy classes — view the full character table.
| maximal subgroup of | #72255279, #72270168, D3×#144388 |
| maximal normal subgroup of | #72255279, #72270168, D3×#144388 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.4*F.5*F.7*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.8*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);