Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | Dic3×#120 |
| Sylow | C3 ⋊ C4×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C2×C3×#120 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | C4×#120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
1224 conjugacy classes of subgroups; only the 289 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 289 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)(13,15);(9,11);(8,9)(10,11)(12,13)(14,15);(2,3)(4,5,6,7)84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×AΣL(1,𝔽16), #10711148, Dic3×B(4,𝔽2), … |
| maximal normal subgroup of | Dic3×AΣL(1,𝔽16), #10711148, Dic3×B(4,𝔽2), … |
| maximal quotient of | Dic3×#386, Dic3×#385, Dic3×#379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15153);