Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| direct product | ΓL(1,𝔽8)×#4207 |
| Fitting | C7×#737 ⋊ C2×C3 |
| derived | C3×C7×C9 ⋊ C2×C32 |
| Hall | ΓL(1,𝔽8)×#737 ⋊ C2 |
| Sylow | C7 ⋊ C3×#4207 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C7×C9 | C2×C32 |
| Frattini | C3 | C3×C3⋊D3×ΓL(1,𝔽8) |
| Fitting | C7×#737 | C2×C3 |
| socle | C3×C7 | C32×C3⋊D3 |
| LCS | G ▷ C3×C7×C9 | |
| UCS | C1 | |
372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 189 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#4207⋊C3, AGL(1,𝔽7)×#4207 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#4207⋊C3, AGL(1,𝔽7)×#4207 |
Past the SmallGroups library coverage — no SmallGroups id.