Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | D3×#382 |
| Fitting | C3×#382 ⋊ C2 |
| Sylow | C3 ⋊ C2×#382 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#382 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C23×D3 | C23 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | C2×#382 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
712 conjugacy classes of subgroups; only the 269 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 269 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(8,10,11,9)(13,17)(16,19);(8,9,11,10)(12,14,15,18)(13,16,17,19);(4,5,6,7)(9,10)(12,13)(14,16)(15,17)(18,19);(2,3)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1286, #10684977, D3×#1295, … |
| maximal normal subgroup of | D3×#1286, #10684977, D3×#1295, … |
| maximal quotient of | D3×#1434, D3×C4⋊#95, #10716414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6799);