Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D3×#532 |
| Fitting | C3×#532 ⋊ C2 |
| Sylow | C3 ⋊ C2×#532 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C3×#532 | C2 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | C2×#532 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2,3)(8,10,11,14)(9,13,12,15);(4,5,6,7);(9,12)(13,15);(5,7)(8,9)(10,13)(11,12)(14,15);(2,3)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1480, D3×#1484, #10761725, … |
| maximal normal subgroup of | D3×#1480, D3×#1484, #10761725, … |
| maximal quotient of | D3×#2285, D3×#2293, D3×#2297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18810);