Finite Group DatabasebetaFinite Group Databasebeta| order | 4200 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | ΓL(1,𝔽8)×#6038 |
| Hall | C7×#6038 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ D4 |
| Hall | C52×C7 ⋊ C3×D4 |
| Sylow | C52 ⋊ D4×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×#6038 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D52×ΓL(1,𝔽8) |
| commutator, Fitting, socle, cosocle | C2×C52×C7 | C22×C3 |
| hypocenter | C52×C7 | C3×D4 |
| LCS | G ▷ C2×C52×C7 ▷ C52×C7 | |
| UCS | C1 ◁ C2 | |
168 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
145 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#87886, AGL(1,𝔽7)×#6038, #72243213 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#87886, AGL(1,𝔽7)×#6038, #72243213 |
Past the SmallGroups library coverage — no SmallGroups id.