Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#5081 |
| 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)(16,21)(18,20);(6,7,9,11)(8,12,13,10)(15,18)(19,20);(4,5)(14,15,16,20,17,18,21,19);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10690106, #10690112, #10696488, … |
| maximal normal subgroup of | #10690106, #10690112, #10696488, … |
| maximal quotient of | C2×#69578, C2×#69620, C2×#69621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15782);