Finite Group DatabasebetaFinite Group Databasebeta| order | 4200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | D5×AGL(1,𝔽5)×ΓL(1,𝔽8) |
| Fitting, socle, derived, Hall | C52×C7 ⋊ C2×C3×C4 |
| Hall | C7×D5×AGL(1,𝔽5) ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C2×C4 |
| Sylow | C52 ⋊ C2×C4×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×D5×AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C7 | C2×C3×C4 |
| prime radical | D52×ΓL(1,𝔽8) | C2 |
| cosocle | C5×C7×D5 | C22×C3 |
| LCS | G ▷ C52×C7 | |
| UCS | C1 | |
136 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 136 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 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)2×ΓL(1,𝔽8), D5×AGL(1,𝔽5)×AGL(1,𝔽7), AGL(1,𝔽5)×ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
| maximal normal subgroup of | AGL(1,𝔽5)2×ΓL(1,𝔽8), D5×AGL(1,𝔽5)×AGL(1,𝔽7), AGL(1,𝔽5)×ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id.