Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×C7⋊#4183 |
| Fitting, Hall | C3×C7×AΣL(1,𝔽27) ⋊ C2 |
| derived | C7×AΣL(1,𝔽27) ⋊ C2×C3 |
| Sylow | C3×AΣL(1,𝔽27) ⋊ D7 |
| Sylow | C7 ⋊ C3×#4183 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C7⋊#4183 |
| commutator, cosocle, hypocenter | C7×AΣL(1,𝔽27) | C2×C3 |
| Frattini | C32 | C3×C3⋊D21 |
| Fitting | C3×C7×AΣL(1,𝔽27) | C2 |
| socle | C32×C7 | #272 |
| LCS | G ▷ C7×AΣL(1,𝔽27) | |
| UCS | C1 ◁ C3 | |
300 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
192 conjugacy classes — view the full character table.
| maximal subgroup of | #72249205, C3×#72249208, C3×D7×#4183 |
| maximal normal subgroup of | C3×#72249208, C3×D7×#4183 |
Past the SmallGroups library coverage — no SmallGroups id.