Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#4530 |
| Sylow | C3 ⋊ C2×#343 |
432 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 205 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 17 points:
(3,4,5)(6,7,8,9)(11,14)(12,16);(4,5)(10,11,12,15,13,14,16,17);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10682309, #10682311, Dic3×#343, … |
| maximal normal subgroup of | #10682309, #10682311, Dic3×#343, … |
| maximal quotient of | C2×#67613, #10684784, #10684787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4137);