Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | A4×#137 |
| unambiguous semidirect product | (C22×#137)⋊C3 |
| Fitting, Sylow | C22×#137 ⋊ C3 |
344 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,4)(6,11)(10,19)(12,16)(13,20);(1,4)(2,3)(5,15,9,7)(6,19,11,10)(8,14,18,17)(12,13,16,20);(5,6)(7,13)(8,16)(9,11)(10,14)(12,18)(15,20)(17,19)44 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#455, A4×#456, A4×#471, … |
| maximal normal subgroup of | A4×#455, A4×#456, A4×#471, … |
| maximal quotient of | A4×#419, A4×#417, A4×#428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18211);