Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| Fitting | C3×#11402 ⋊ C2 |
| Sylow | C3 ⋊ #141754 |
1016 conjugacy classes of subgroups; only the 225 normal subgroups are shown.
This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 225 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,6,8,7,9,10,11);(5,9)(8,11)(12,14,21,19,17,22,25,20,18,23,16,26,15,24,27,13);(2,3)(5,8)(6,10)(9,11)(12,13,15,19,18,20,17,26)(14,16,22,25,23,21,24,27)Polycyclic presentation:
2^9,3|a^2=d,b^2=dfhi,c^2=e,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bf,c^a=ce,c^b=cg,d^b=di,e^a=eg,f^a=fh,f^b=fi,h^a=hi,j^a=j^2180 conjugacy classes — view the full character table.
| maximal subgroup of | #72225893, #72226204 |
| maximal normal subgroup of | #72225893, #72226204 |
| maximal quotient of | #72226204 |
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, 2, F.4*F.6*F.8*F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);