Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#4765 |
| Sylow | C3 ⋊ C2×#489 |
436 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 141 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)(6,8)(7,10)(9,13)(11,12)(14,15)(16,21)(17,18)(19,20);(1,2)(6,8,9,13)(7,10,11,12)(15,19)(16,20)(18,21);(4,5)(6,7,9,11)(8,12,13,10)(15,18)(19,21);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4765, #10803252, #10803296, … |
| maximal normal subgroup of | C4×#4765, #10803252, #10803296, … |
| maximal quotient of | C2×#68849, C2×#68854, C2×#68855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11311);