Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | AGL(1,𝔽7)×#585 |
| Fitting | C7×#585 ⋊ C2×C3 |
| Hall | D7×#585 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×#585 |
| Sylow | C7 ⋊ C2×C3×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C7 | C26×C3 |
| Frattini | C2 | C25×AGL(1,𝔽7) |
| Fitting | C7×#585 | C2×C3 |
| hypercenter | #585 | AGL(1,𝔽7) |
| hypocenter | C7 | C2×C3×#585 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ #585 | |
11912 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
238 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5950, AGL(1,𝔽7)×#3585, AGL(1,𝔽7)×#3025, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#5950, AGL(1,𝔽7)×#3585, AGL(1,𝔽7)×#3025, … |
Past the SmallGroups library coverage — no SmallGroups id.