Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | ΓL(1,𝔽8)×#4181 |
| Fitting, derived | C7×C32⋊C9 ⋊ C2×C3 |
| Hall | ΓL(1,𝔽8)×C32⋊C9 ⋊ C2 |
| Sylow | C7 ⋊ C3×#4181 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽8)×C3⋊D9 |
| commutator, Fitting, cosocle, hypocenter | C7×C32⋊C9 | C2×C3 |
| Frattini | C32 | C3⋊D3×ΓL(1,𝔽8) |
| socle | C32×C7 | C3×C3⋊D3 |
| LCS | G ▷ C7×C32⋊C9 | |
| UCS | C1 ◁ C3 | |
312 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#92302, AGL(1,𝔽7)×#4181 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#92302, AGL(1,𝔽7)×#4181 |
Past the SmallGroups library coverage — no SmallGroups id.