Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | D3×#543 |
| Fitting | C3×#543 ⋊ C2 |
| Sylow | C3 ⋊ C2×#543 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#543 | C2 |
| prime radical | C24×D3 | C22 |
| hypercenter | #543 | D3 |
| hypocenter | C3 | C2×#543 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #543 | |
1296 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 517 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) = 48 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(5,9)(8,10);(6,11)(8,10)(12,14,15,19)(13,16,17,18);(4,6)(5,8)(7,11)(9,10);(4,5)(6,10)(7,9)(8,11)(12,13,15,17)(14,18,19,16);(2,3)66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1596, D3×#1602, D3×#1608, … |
| maximal normal subgroup of | D3×#1596, D3×#1602, D3×#1608, … |
| maximal quotient of | D3×#2315, D3×#2329, #10954338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19154);