Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×D4×A4 |
| unambiguous semidirect product | (C22×C4×D4)⋊C3 |
| Fitting, Sylow | C22×C4×D4 ⋊ C3 |
578 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 12 points:
(6,8)(10,11,12);(5,6)(7,8)(9,12)(10,11);(1,2,3,4)80 conjugacy classes — view the full character table.
| maximal subgroup of | A4×D4⋊C8, C8×D4×A4, A4×#435, … |
| maximal normal subgroup of | A4×D4⋊C8, C8×D4×A4, A4×#435, … |
| maximal quotient of | C4×A4×C22⋊C4, C4×A4×C4⋊C4, A4×#385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18149);