Finite Group DatabasebetaFinite Group Databasebeta| order | 4200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C5 |
| direct product | C2×ΓL(1,𝔽8)⋊#1044 |
| Fitting, socle | C2×C52×C7 ⋊ C3×C4 |
| derived, Hall | C52×C7 ⋊ C2×C3×C4 |
| Hall | C2×C7⋊#1044 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C2×C4 |
| Sylow | C52 ⋊ C2×ΓL(1,𝔽8)⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ C2×#1044 |
| Sylow | C7 ⋊ C2×C3×#1044 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)⋊#1044 |
| commutator, hypocenter | C52×C7 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C52×C7 | C3×C4 |
| prime radical | C2×ΓL(1,𝔽8)×C5⋊D5 | C2 |
| cosocle | C7×C5⋊D5 | C22×C3 |
| LCS | G ▷ C52×C7 | |
| UCS | C1 ◁ C2 | |
168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | #72243125, #72243131, C2×AGL(1,𝔽7)×#1044, … |
| maximal normal subgroup of | #72243125, #72243131, C2×AGL(1,𝔽7)×#1044, … |
| maximal quotient of | #72243199 |
Past the SmallGroups library coverage — no SmallGroups id.