Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#11714267 |
| derived | C22⋊(C22⋊A4) ⋊ C2×C4 |
4273 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 4273 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 24 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,8,7)(4,12,11)(5,14,13)(9,16,18);(3,12,5,17,8,11,14,9)(4,15,18,7,10,13,16,6)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215860, C2×#72215875, C2×#72215879, … |
| maximal normal subgroup of | C2×#72215860, C2×#72215875, C2×#72215879, … |
| maximal quotient of | #72233209, C2×#72215789, C4×#11714267, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,8,7)(4,12,11)(5,14,13)(9,16,18), (3,12,5,17,8,11,14,9)(4,15,18,7,10,13,16,6));