Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5201 |
| Fitting | C2×C3×AGL(1,ℤ/8) ⋊ C2 |
| Sylow | C3 ⋊ C2×#454 |
728 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 728 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)(7,13)(8,16)(9,10)(14,15)(17,18)(19,20);(1,2)(6,8,11,18)(7,12,15,21)(9,17,10,16)(13,20,14,19);(4,5)(6,7)(8,12)(9,13)(10,14)(11,15)(16,19)(17,20)(18,21);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5201, #10801086, #10801123, … |
| maximal normal subgroup of | C4×#5201, #10801086, #10801123, … |
| maximal quotient of | C2×#70970, C2×#70978, C2×#71170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16536);