Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | D3×#386 |
| Fitting | C3×#386 ⋊ C2 |
| Sylow | C3 ⋊ C2×#386 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#386 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×D3×D4 | C2 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | C2×#386 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(13,15);(4,6,7,5)(9,11);(4,5,7,6)(8,9,10,11)(12,13)(14,15);(2,3)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1390, #10711424, D3×#1790, … |
| maximal normal subgroup of | D3×#1390, #10711424, D3×#1790, … |
| maximal quotient of | D3×C22⋊#95, D3×#1436, D3×#1437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7449);