Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×C5×D3 (order 240) · Inn(G) = D3 · Out(G) = C23×C5 |
| homology | Abelianization (H1) = C3×C4×C11 · Schur multiplier (H2) = C1 |
| direct product | C3×C11×Dic3 |
| unambiguous semidirect product | C3⋊(C3×C4×C11) |
| derived | C3 ⋊ C3×C4×C11 |
| Hall | C32×C11 ⋊ C4 |
| Sylow | C32 ⋊ C4×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C11 | D3 |
| commutator, hypocenter | C3 | C3×C4×C11 |
| Frattini | C2 | C3×D3×C11 |
| Fitting, socle, prime radical | C2×C32×C11 | C2 |
| cosocle | C2×C3 | C2×C3×C11 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C3×C11 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 132 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3)(5,6)(7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21);(4,5,6)198 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic3×D11, C3×D33⋊C4, C3×#63141, … |
| maximal normal subgroup of | C3×Dic3×D11, C3×D33⋊C4, C3×#63141, … |
| maximal quotient of | C3×C11×C3⋊C8, C2×C3×C11×Dic3, C3×C11×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 12);