Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#4791 |
| Sylow | C3 ⋊ C2×#475 |
416 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 416 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)(6,9,10,8)(7,11,12,13)(15,18)(16,21)(19,20);(1,2)(6,8)(7,11)(9,10)(12,13)(14,15,17,20)(16,18,21,19);(4,5)(6,7,10,12)(8,11,9,13);(1,2)66 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4791, #10802768, #10802841, … |
| maximal normal subgroup of | C4×#4791, #10802768, #10802841, … |
| maximal quotient of | C2×#70221, C2×#70237, C2×#70238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11351);