Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D3×#396 |
| Fitting | C3×#396 ⋊ C2 |
| Sylow | C3 ⋊ C2×#396 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#396 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #396 | D3 |
| hypocenter | C3 | C2×#396 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #396 | |
908 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 203 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) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(5,7)(9,11)(12,13)(14,15);(8,9)(10,11)(12,13,14,15);(4,5,6,7)(13,15);(2,3)66 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×#396, D3×#2392, D3×#2396, … |
| maximal normal subgroup of | C2×D3×#396, D3×#2392, D3×#2396, … |
| maximal quotient of | D3×#1439, D3×#1441, D3×#1443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9989);