Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#5200 |
| Sylow | C3 ⋊ C2×#471 |
468 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 141 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) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5)(7,13)(8,17)(9,10)(12,20)(14,15)(18,19);(1,2)(6,8)(7,12)(9,18)(10,17)(11,19)(13,21)(14,16)(15,20);(4,5)(6,7,11,15)(8,16,19,21)(9,13,10,14)(12,17,20,18);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5200, #10802057, #10802107, … |
| maximal normal subgroup of | C4×#5200, #10802057, #10802107, … |
| maximal quotient of | C2×#71004, C2×#71132, C2×#71007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16517);