Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#4856 |
| Sylow | C3 ⋊ C2×#501 |
384 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5)(6,8,9,13)(7,10,11,12)(15,19)(16,20)(18,21);(14,15,16,19,17,18,20,21);(4,5)(6,7,9,11)(8,12,13,10)(15,18)(19,21);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4856, #10803998, #10804006, … |
| maximal normal subgroup of | C4×#4856, #10803998, #10804006, … |
| maximal quotient of | C2×#70556, C2×#70561, C2×#68860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11685);