Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | A4×#117 |
| unambiguous semidirect product | (C22×#117)⋊C3 |
| Fitting, Sylow | C22×#117 ⋊ C3 |
440 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 16 points:
(6,7,8)(9,15,12,11)(10,16,13,14);(5,8)(6,7)(10,13)(14,16);(1,4,3,2)(9,10)(11,14)(12,13)(15,16)80 conjugacy classes — view the full character table.
| maximal subgroup of | A4×H(3,ℤ/4), A4×#339, A4×#344, … |
| maximal normal subgroup of | A4×H(3,ℤ/4), A4×#339, A4×#344, … |
| maximal quotient of | A4×#376, C4×A4×C22⋊C4, C4×A4×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18148);