Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | A4×SD32 |
| unambiguous semidirect product | (C22×SD32)⋊C3 |
| Fitting, Sylow | C22×SD32 ⋊ C3 |
110 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 20 points:
(2,3,4)(6,10)(7,14)(8,15)(11,12)(13,19)(16,20)(17,18);(1,4)(2,3)(5,13,9,6)(7,19,16,10)(8,12,15,17)(11,14,18,20)44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×SD32, A4×#508, A4×#509, … |
| maximal normal subgroup of | C2×A4×SD32, A4×#508, A4×#509, … |
| maximal quotient of | A4×#357, A4×#358, A4×#367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5854);