Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of D33 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic33 |
| unambiguous semidirect product | C3⋊(C3×Dic11) |
| unambiguous semidirect product | C11⋊(C3×Dic3) |
| derived | C3×C11 ⋊ C3×C4 |
| Hall | C32×C11 ⋊ C4 |
| Sylow | C11 ⋊ C3×Dic3 |
| Sylow | C32 ⋊ Dic11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D33 |
| commutator, hypocenter | C3×C11 | C3×C4 |
| Frattini | C2 | C3×D33 |
| Fitting, socle, prime radical | C2×C32×C11 | C2 |
| cosocle | C2×C3×C11 | C2×C3 |
| LCS | G ▷ C3×C11 | |
| UCS | C1 ◁ C2×C3 | |
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:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,14)(16,17)(18,21,20,19);(1,2,4,6,8,10,11,9,7,5,3)(15,16,17)108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic3×D11, C3×D3×Dic11, C3×#63140, … |
| maximal normal subgroup of | C3×Dic3×D11, C3×D3×Dic11, C3×#63140, … |
| maximal quotient of | C3×C3⋊(C11⋊C8), C2×C3×Dic33, C3×Dic99, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 13);