Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of D11 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C32×D11 |
| unambiguous semidirect product | C11⋊(C22×C32) |
| unambiguous semidirect product | (C2×C11)⋊(C2×C32) |
| Fitting, socle | C2×C32×C11 ⋊ C2 |
| derived, Sylow | C11 ⋊ C22×C32 |
| Hall | C32×C11 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D11 |
| commutator, cosocle, hypocenter | C11 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C11 | C2 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2×C32 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 198 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2)(3,4,5)(10,11)(12,13)(14,15)(16,17)(18,19);(1,2)(6,7,8)(9,10,12,14,16,18,19,17,15,13,11)126 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic3×D11, C3×#63140, C3×#63141, … |
| maximal normal subgroup of | C3×Dic3×D11, C3×#63140, C3×#63141, … |
| maximal quotient of | C32×Dic22, C32×C4×D11, C32×D44, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 25);