Finite Group DatabasebetaFinite Group Databasebeta| order | 2160 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C5×C22⋊C9 ⋊ C3×C4 |
| Sylow | C5 ⋊ #1094⋊C4 |
144 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 90 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,4,3,2)(6,7,8,9)(11,15,16,13,14,18)(12,17);(2,3,4)(5,6,9,7,8)(10,16,13,12,14,15,17,11,18)Polycyclic presentation:
2,3,2,3,5,3,2^2|a^2=c,d^3=f,d^a=d^2f^2,d^b=df^2,e^a=e^3,e^c=e^4,f^a=f^2,g^a=gh,g^d=h,h^d=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #72247686, AGL(1,𝔽5)×((C22⋊C9)⋊C2)⋊C3 |
| maximal normal subgroup of | AGL(1,𝔽5)×((C22⋊C9)⋊C2)⋊C3 |
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");;
gap> coll := SingleCollector(F, [2, 3, 2, 3, 5, 3, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^3);;
gap> SetConjugate(coll, 5, 3, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);