Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#12687 ⋊ C2 |
| Sylow | C3 ⋊ #144388 |
2844 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 2844 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:
(1,3)(2,5,6,7)(4,8)(9,11,13,20)(10,15)(12,19)(14,22)(16,23,17,18)(21,24)(26,27);(5,7)(10,14)(19,24)(25,26,27);(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,14,21,17)(11,18,19,22,20,23,24,15)Polycyclic presentation:
2^9,3|a^2=e,c^2=di,e^2=i,b^a=bef,c^a=cfgi,c^b=cd,d^a=dh,e^b=eg,e^c=egi,e^d=ei,f^a=fg,f^b=fg,f^c=fhi,f^d=fi,f^e=fi,g^a=gi,g^c=gi,h^a=hi,h^c=hi,j^c=j^284 conjugacy classes — view the full character table.
| maximal subgroup of | #72255279, D3×#144388, #72270550 |
| maximal normal subgroup of | #72255279, D3×#144388, #72270550 |
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.5);;
gap> SetPower(coll, 3, F.4*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
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);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, 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);