Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C32×D3 (order 216) · Inn(G) = D3 · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C2×C7×C9 · Schur multiplier (H2) = C1 |
| direct product | D3×C7×C9 |
| unambiguous semidirect product | C3⋊(C2×C7×C9) |
| Fitting, Hall | C3×C7×C9 ⋊ C2 |
| derived | C3 ⋊ C2×C7×C9 |
| Sylow | C3×C9 ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C9 | D3 |
| commutator, hypocenter | C3 | C2×C7×C9 |
| Frattini | C3 | C3×D3×C7 |
| Fitting | C3×C7×C9 | C2 |
| socle | C32×C7 | C2×C3 |
| prime radical | C3×D3×C7 | C3 |
| cosocle | C32 | C2×C3×C7 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C7×C9 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,3)(4,5,6,7,8,9,10)(11,18,17,13,12,19,16,15,14);(1,2,3)189 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C9×D7, D3×C7×D9, D3×D63, … |
| maximal normal subgroup of | D3×C9×D7, D3×C7×D9, D3×D63, … |
| maximal quotient of | C7×C9×Dic3, C2×D3×C7×C9, C7×C9×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 33);