Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#4525 |
| Sylow | C3 ⋊ C2×#323 |
456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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 — 3 generators on 21 points:
(3,4,5)(6,7)(8,10)(9,11)(12,13)(14,16)(15,19)(17,18)(20,21);(4,5)(7,10,11,13)(8,12)(14,15,17,19,18,20,16,21);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10682145, #10682146, #10682306, … |
| maximal normal subgroup of | #10682145, #10682146, #10682306, … |
| maximal quotient of | C2×#67639, C2×#67642, #10683139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4114);