Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | D3×#394 |
| Fitting | C3×#394 ⋊ C2 |
| Sylow | C3 ⋊ C2×#394 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#394 | C2 |
| prime radical | C22×D3×D4 | C2 |
| hypercenter | #394 | D3 |
| hypocenter | C3 | C2×#394 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #394 | |
1252 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)(5,7)(9,11)(12,14,15,13);(8,9)(10,11)(12,13)(14,15);(4,5)(6,7)(13,14);(2,3)66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1345, #10705867, D3×#1597, … |
| maximal normal subgroup of | D3×#1345, #10705867, D3×#1597, … |
| maximal quotient of | D3×#1438, D3×#1439, D3×#1440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 9681);