Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
| direct product | Dic3×#95 |
| Sylow | C3 ⋊ C4×#95 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C22×D3 |
| commutator | C2×C3 | C43 |
| Fitting | C2×C3×#95 | C2 |
| socle, prime radical, cosocle | C24×C3 | C23 |
| hypercenter | C2×#95 | D3 |
| hypocenter | C3 | C4×#95 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C2×#95 | |
752 conjugacy classes of subgroups; only the 325 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 325 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) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(9,10,11,12)(14,16)(17,18,19);(5,6,7,8)(13,14)(15,16);(1,2,3,4)(18,19)120 conjugacy classes — view the full character table.
| maximal subgroup of | #10686060, Dic3×#327, #10686065, … |
| maximal normal subgroup of | #10686060, Dic3×#327, #10686065, … |
| maximal quotient of | #10683106, Dic3×#336, #10700335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1976);