Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×#4183 |
| Fitting, derived | C7×AΣL(1,𝔽27) ⋊ C2×C3 |
| Hall | ΓL(1,𝔽8)×AΣL(1,𝔽27) ⋊ C2 |
| Sylow | C7 ⋊ C3×#4183 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×AΣL(1,𝔽27) | C2×C3 |
| Frattini | C32 | C3⋊D3×ΓL(1,𝔽8) |
| socle | C3×C7 | C3×#272 |
| LCS | G ▷ C7×AΣL(1,𝔽27) | |
| UCS | C1 | |
300 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 63 |
65 conjugacy classes — view the full character table.
| maximal subgroup of | C3×ΓL(1,𝔽8)×#4183, ΓL(1,𝔽8)×#4183⋊C3, AGL(1,𝔽7)×#4183 |
| maximal normal subgroup of | C3×ΓL(1,𝔽8)×#4183, ΓL(1,𝔽8)×#4183⋊C3, AGL(1,𝔽7)×#4183 |
| maximal quotient of | C3×ΓL(1,𝔽8)×#4183 |
Past the SmallGroups library coverage — no SmallGroups id.