Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×C4×C22⋊(C22⋊A4) |
| Fitting, Sylow | C27×C4 ⋊ C3 |
| derived | C26 ⋊ C2×C3×C4 |
268982 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 268982 conjugacy classes of subgroups — too many to draw. Only its 360 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) = 48 |
A minimal faithful permutation representation — 4 generators on 18 points:
(4,6,5)(8,10,9)(12,13,14)(15,17,18,16);(1,2)(11,14)(12,13)(15,16,18,17);(7,10)(8,9);(3,6)(4,5)192 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#85645, C2×C4×#87588 |
| maximal normal subgroup of | C2×C4×#85645, C2×C4×#87588 |
| maximal quotient of | C2×C4×C22⋊#997 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,6,5)(8,10,9)(12,13,14)(15,17,18,16), (1,2)(11,14)(12,13)(15,16,18,17), (7,10)(8,9), (3,6)(4,5));