Finite Group DatabasebetaFinite Group Databasebeta| order | 8400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | ΓL(1,𝔽8)×#87832 |
| Hall | C7×#87832 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C4⋊C4 |
| Hall | C52×C7 ⋊ C3×C4⋊C4 |
| Sylow | C52 ⋊ C4⋊C4×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#87832 |
272 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
110 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#11717375, AGL(1,𝔽7)×#87832, #72243223 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#11717375, AGL(1,𝔽7)×#87832, #72243223 |
Past the SmallGroups library coverage — no SmallGroups id.