Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×#936 |
| Sylow | C3 ⋊ C22×#124 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting, prime radical | C23×C3×D4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C23×D4 | D3 |
| hypocenter | C3 | C22×#124 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
1880 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 645 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) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(3,4)(5,6,7)(9,11)(12,14)(13,16)(15,18)(17,19);(12,13,15,17)(14,19,18,16);(6,7)(8,9,10,11)(14,18)(16,19);(3,4);(1,2)96 conjugacy classes — view the full character table.
| maximal subgroup of | #10687799, #10701727, #10701733, … |
| maximal normal subgroup of | #10687799, #10701727, #10701733, … |
| maximal quotient of | C22×#4977, C22×#4945, C22×#5221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19807);