Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×D7×AGL(1,𝔽7) |
| Fitting | C4×C72 ⋊ C22×C3 |
| derived, Sylow | C72 ⋊ C22×C3×C4 |
| Hall | C4×D72 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D7×AGL(1,𝔽7) |
| commutator, hypocenter | C72 | C22×C3×C4 |
| Frattini | C2 | C2×D7×AGL(1,𝔽7) |
| Fitting | C4×C72 | C22×C3 |
| socle, cosocle | C2×C72 | C23×C3 |
| prime radical | C2×D7×AGL(1,𝔽7) | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C4 | |
224 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | #72245018, C4×AGL(1,𝔽7)2, #72245794, … |
| maximal normal subgroup of | #72245018, C4×AGL(1,𝔽7)2, #72245794, … |
| maximal quotient of | #72245767, D7×AGL(1,𝔽5)×AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id.