Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4×C7 |
| direct product | C42×#64425 |
| Fitting | C42×C72 ⋊ C2×C3 |
| derived, Sylow | C72 ⋊ C2×C3×C42 |
| Hall | C42×C7⋊D7 ⋊ C3 |
| Hall | #3862 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | #64425 |
| commutator, hypocenter | C72 | C2×C3×C42 |
| Frattini | C22 | C22×#64425 |
| Fitting | C42×C72 | C2×C3 |
| socle, cosocle | C22×C72 | C23×C3 |
| prime radical | C22×#64425 | C22 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C42 | |
540 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
224 conjugacy classes — view the full character table.
| maximal subgroup of | #127×#64425, C42×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72245996, … |
| maximal normal subgroup of | #127×#64425, C42×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72245996, … |
Past the SmallGroups library coverage — no SmallGroups id.