Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5205 |
| Fitting | C2×C3×#137 ⋊ C2 |
| Sylow | C3 ⋊ C2×#456 |
708 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 149 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)(6,9,12,13)(7,14,16,21)(8,17,18,15)(10,20,11,19);(1,2)(6,8,12,18)(7,13,16,9)(10,14,11,21)(15,20,17,19);(4,5)(6,7)(8,13)(9,18)(10,17)(11,15)(12,16)(14,19)(20,21);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5205, #10801097, #10801134, … |
| maximal normal subgroup of | C4×#5205, #10801097, #10801134, … |
| maximal quotient of | C2×#70974, C2×#70982, C2×#71166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16571);