Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C3×C7×#4183 |
| Fitting, Hall | C3×C7×AΣL(1,𝔽27) ⋊ C2 |
| derived | AΣL(1,𝔽27) ⋊ C2×C3×C7 |
| Sylow | C3×AΣL(1,𝔽27) ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C7 | #4183 |
| commutator, cosocle, hypocenter | AΣL(1,𝔽27) | C2×C3×C7 |
| Frattini | C32 | C3×C7×C3⋊D3 |
| Fitting | C3×C7×AΣL(1,𝔽27) | C2 |
| socle | C32×C7 | #272 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C3×C7 | |
300 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
273 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×#13091120, C3×ΓL(1,𝔽8)×#4183, C3×D7×#4183 |
| maximal normal subgroup of | C3×ΓL(1,𝔽8)×#4183, C3×D7×#4183 |
Past the SmallGroups library coverage — no SmallGroups id.