Finite Group DatabasebetaFinite Group Databasebeta| order | 4200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×D5×C5⋊AGL(1,𝔽7) |
| Fitting, socle | C2×C52×C7 ⋊ C22×C3 |
| derived, Hall | C52×C7 ⋊ C23×C3 |
| Hall | C2×D5×D35 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C23 |
| Sylow | C52 ⋊ C22×AGL(1,𝔽7) |
| Hall | ΓL(1,𝔽8) ⋊ C2×D52 |
| Sylow | C7 ⋊ C2×C3×D52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×C5⋊AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C52×C7 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C52×C7 | C22×C3 |
| LCS | G ▷ C52×C7 | |
| UCS | C1 ◁ C2 | |
296 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D52×AGL(1,𝔽7), #72243211, #72243213 |
| maximal normal subgroup of | C2×D52×AGL(1,𝔽7), #72243211, #72243213 |
| maximal quotient of | #72243211 |
Past the SmallGroups library coverage — no SmallGroups id.