Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C22×D7×AGL(1,𝔽7) |
| Fitting, socle | C22×C72 ⋊ C22×C3 |
| derived, Sylow | C72 ⋊ C24×C3 |
| Hall | C22×D72 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C72 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72 | C22×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C22 | |
552 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 187 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 | C2×#72222021, C22×AGL(1,𝔽7)2, #72245767, … |
| maximal normal subgroup of | C2×#72222021, C22×AGL(1,𝔽7)2, #72245767, … |
| maximal quotient of | #72245018, C2×#72222021, #72245780, … |
Past the SmallGroups library coverage — no SmallGroups id.