Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D3×#546 |
| Fitting | C3×#546 ⋊ C2 |
| Sylow | C3 ⋊ C2×#546 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#546 | C2 |
| hypercenter | #546 | D3 |
| hypocenter | C3 | C2×#546 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #546 | |
1900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2,3)(10,15)(12,14);(5,7);(8,10)(9,12)(11,15)(13,14);(4,5)(6,7)(8,9,11,13)(10,14,15,12);(2,3)75 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3264, D3×#3266, D3×#3267, … |
| maximal normal subgroup of | D3×#3264, D3×#3266, D3×#3267, … |
| maximal quotient of | D3×D4×C22⋊C4, D3×#2339, D3×#2342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19237);