Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#4760 |
| Fitting | C2×C3×#102 ⋊ C2 |
| Sylow | C3 ⋊ C2×#486 |
528 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 157 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) = 192 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5)(6,9,10,8)(7,12,13,11)(14,15)(16,21)(17,19)(18,20);(1,2)(6,8)(7,11)(9,10)(12,13)(15,18)(16,20)(19,21);(4,5)(6,7,10,13)(8,12,9,11);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4760, #10803151, #10803194, … |
| maximal normal subgroup of | C4×#4760, #10803151, #10803194, … |
| maximal quotient of | C2×#68847, C2×#70186, C2×#68852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11309);