Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C4×B(4,𝔽2)⋊C3 |
| Fitting, Sylow | C2×C4×B(4,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C22×C3×C4 |
2728 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 2728 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 14 points:
(3,4,5,6)(8,12,9)(11,14,13);(7,14)(8,13,11,9)(10,12);(1,2)(8,11)(9,13)(12,14)128 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#11714171, C2×C4×#73026, C4×#60399⋊C3, … |
| maximal normal subgroup of | C4×#11714171, C2×C4×#73026, C4×#60399⋊C3, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5,6)(8,12,9)(11,14,13), (7,14)(8,13,11,9)(10,12), (1,2)(8,11)(9,13)(12,14));