Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of D3×D11 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽11) (order 1320) · Inn(G) = D3×D11 · Out(G) = C2×C5 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D3×D11 |
| unambiguous semidirect product | (D3×C11)⋊(C2×C3) |
| unambiguous semidirect product | (C3×D11)⋊(C2×C3) |
| unambiguous semidirect product | D33⋊(C2×C3) |
| Fitting, socle, Hall | C32×C11 ⋊ C22 |
| derived | C3×C11 ⋊ C22×C3 |
| Sylow | C11 ⋊ C2×C3×D3 |
| Sylow | C32 ⋊ C2×D11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×D11 |
| commutator, cosocle, hypocenter | C3×C11 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C32×C11 | C22 |
| LCS | G ▷ C3×C11 | |
| UCS | C1 ◁ C3 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 66 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3)(4,5,6)(8,9)(10,11)(12,13)(14,15)(16,17);(5,6)(7,8,10,12,14,16,17,15,13,11,9)63 conjugacy classes — view the full character table.
| maximal subgroup of | D32×D11, C2×C3×D3×D11, C3×D9×D11, … |
| maximal normal subgroup of | D32×D11, C2×C3×D3×D11, D3×C9×D11, … |
| maximal quotient of | C3×Dic3×D11, C3×D3×Dic11, C3×D33⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 19);