Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#12693 ⋊ C2 |
| Sylow | C3 ⋊ #512119 |
2960 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 2960 conjugacy classes of subgroups — too many to draw. Only its 221 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 — 4 generators on 27 points:
(1,2,3)(5,9)(8,11)(15,23)(19,25)(20,26)(21,27);(2,3)(4,6,7,10)(5,8,9,11)(13,18)(17,24)(19,25)(21,27);(4,5,7,9)(6,8,10,11)(12,14)(13,17)(15,21,23,27)(16,22)(18,24)(19,20,25,26);(12,13)(14,20,22,26)(15,19)(16,18)(17,21,24,27)(23,25)Polycyclic presentation:
2^9,3|a^2=h,b^2=e,c^2=g,f^2=i,b^a=bf,c^a=chi,c^b=ceg,d^a=dh,d^b=dg,e^a=eh,f^b=fhi,f^c=fh,f^d=fi,f^e=fi,h^b=hi,j^c=j^293 conjugacy classes — view the full character table.
| maximal subgroup of | #72269981, #72270758, D3×#512119 |
| maximal normal subgroup of | #72269981, #72270758, D3×#512119 |
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.8);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);