Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C32 |
| direct product | C7×#92300 |
| Fitting, Hall | C7×C32⋊H(3,𝔽3) ⋊ C2 |
| derived | C3×H(3,𝔽3) ⋊ C2×C3×C7 |
| Sylow | C32⋊H(3,𝔽3) ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C7 | C3⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C3×H(3,𝔽3) | C2×C3×C7 |
| Frattini | C32 | C3×C7×C3⋊D3 |
| Fitting | C7×C32⋊H(3,𝔽3) | C2 |
| socle | C32×C7 | C3×C3⋊D3 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C3×C7 | |
384 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 46 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, ΓL(1,𝔽8)×#92300, D7×#92300 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#92300, D7×#92300 |
Past the SmallGroups library coverage — no SmallGroups id.