Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#12696 ⋊ C2 |
| Sylow | C3 ⋊ #133356 |
2332 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 2332 conjugacy classes of subgroups — too many to draw. Only its 129 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)(4,6)(5,8,9,11)(7,10)(12,14)(13,18,20,27)(15,23)(16,22)(17,25,26,19)(21,24);(1,2,3)(4,5)(6,8,10,11)(7,9)(12,14,21,23)(13,17,19,27)(15,24,16,22)(18,20,26,25);(4,5)(6,8,10,11)(7,9)(12,13)(14,18,22,27)(15,19,21,25)(16,20)(17,24)(23,26)Polycyclic presentation:
2^9,3|a^2=d,b^2=ehi,c^2=df,b^a=bde,c^a=cf,c^b=cdef,d^b=dh,d^c=dg,e^a=ehi,e^b=ei,e^c=egi,e^d=ei,f^a=fg,f^b=fghi,f^c=fg,f^e=fi,h^a=hi,h^b=hi,h^c=hi,j^c=j^278 conjugacy classes — view the full character table.
| maximal subgroup of | #72253715, D3×#133356, #72254946 |
| maximal normal subgroup of | #72253715, D3×#133356, #72254946 |
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.5*F.8*F.9);;
gap> SetPower(coll, 3, F.4*F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*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);