Finite Group DatabasebetaFinite Group Databasebeta| order | 6480 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C22×C5×3−1+2 ⋊ Dic3 |
| derived | C5×C9×A4 ⋊ C3×C4 |
| Sylow | C5 ⋊ A4⋊(3−1+2⋊C4) |
404 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,4,3,2)(6,7,8,9)(11,15,16,13,14,18)(12,17);(5,6,9,7,8)(10,16,13,12,14,15,17,11,18);(2,3,4)Polycyclic presentation:
2,3,2,3^2,5,3,2^2|a^2=c,e^3=g,d^a=d^2,e^a=e^2g^2,e^b=eg^2,f^a=f^3,f^c=f^4,g^a=g^2,h^a=hi,h^d=i,i^d=hi106 conjugacy classes — view the full character table.
| maximal subgroup of | S4×3−1+2⋊AGL(1,𝔽5), AGL(1,ℤ/9)×A4⋊AGL(1,𝔽5), AGL(1,𝔽5)×A4⋊AGL(1,ℤ/9) |
| maximal normal subgroup of | S4×3−1+2⋊AGL(1,𝔽5), AGL(1,ℤ/9)×A4⋊AGL(1,𝔽5), AGL(1,𝔽5)×A4⋊AGL(1,ℤ/9) |
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");;
gap> coll := SingleCollector(F, [2, 3, 2, 3, 3, 5, 3, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^3);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);