Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | Dic3×#121 |
| Sylow | C3 ⋊ C4×#121 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting, prime radical | C2×C3×#121 | C2 |
| socle, cosocle | C23×C3 | C24 |
| hypercenter | C2×#121 | D3 |
| hypocenter | C3 | C4×#121 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(9,11);(12,13,14,15);(8,9)(10,11)(13,15);(2,3)(4,5,6,7)84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#327, #10686073, #10686083, … |
| maximal normal subgroup of | Dic3×#327, #10686073, #10686083, … |
| maximal quotient of | Dic3×#386, Dic3×#388, Dic3×#390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15208);