Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | D3×#384 |
| Fitting | C3×#384 ⋊ C2 |
| Sylow | C3 ⋊ C2×#384 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#384 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C23×D3 | C23 |
| hypercenter | #384 | D3 |
| hypocenter | C3 | C2×#384 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #384 | |
888 conjugacy classes of subgroups; only the 365 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 365 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(12,13,14,15);(8,9,10,11);(4,5,6,7)(9,11)(13,15);(2,3)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1296, #10685279, #10685289, … |
| maximal normal subgroup of | D3×#1296, #10685279, #10685289, … |
| maximal quotient of | D3×C4⋊#95, #10716410, Dic3×#384, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6933);