Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #144415 |
1884 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 1884 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(2,3)(4,7)(5,6)(8,10)(9,11)(12,15,16,23)(13,19,26,18)(14,17,24,25)(20,27,21,22);(1,2,3)(4,6)(5,7)(8,11)(9,10)(12,14)(13,18)(15,16)(17,23)(19,21)(20,22)(24,25)(26,27);(4,5,8,9)(6,10)(7,11)(12,13)(14,22,23,18)(15,19,24,27)(16,21)(17,20)(25,26)Polycyclic presentation:
2^9,3|a^2=d,c^2=defhi,f^2=i,b^a=bde,c^a=ceg,c^b=cdefi,d^b=dg,d^c=dg,e^a=eg,e^b=eg,e^c=eh,f^a=fgh,f^c=fghi,f^e=fi,h^b=hi,h^c=hi,j^c=j^284 conjugacy classes — view the full character table.
| maximal subgroup of | #72253601, D3×#144415, #72255279 |
| maximal normal subgroup of | #72253601, D3×#144415, #72255279 |
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.6*F.8*F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);