Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | D3×#381 |
| Fitting | C3×#381 ⋊ C2 |
| Sylow | C3 ⋊ C2×#381 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#381 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×D3 | C22 |
| hypercenter | #381 | D3 |
| hypocenter | C3 | C2×#381 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #381 | |
1144 conjugacy classes of subgroups; only the 269 normal subgroups are shown.
This group has 1144 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) = 96 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(9,11)(12,14,15,18)(13,17,16,19);(13,16)(17,19);(4,5,6,7)(8,9)(10,11)(12,13)(14,17)(15,16)(18,19);(2,3)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1284, #10684913, D3×#1306, … |
| maximal normal subgroup of | D3×#1284, #10684913, D3×#1306, … |
| maximal quotient of | D3×C22⋊#95, D3×#1434, #10716396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6694);