Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C24×C33 |
| direct product | C3×AGL(1,𝔽7)×C22⋊A4 |
| Fitting, socle | C24×C3×C7 ⋊ C2×C32 |
| derived | C24×C7 ⋊ C2×C33 |
| Hall | C24×D7 ⋊ C33 |
| Sylow | C7 ⋊ C2×C32×C22⋊A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | AGL(1,𝔽7)×C22⋊A4 |
| commutator, cosocle, hypocenter | C24×C7 | C2×C33 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C7 | C2×C32 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C3 | |
2224 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 2224 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 252 |
168 conjugacy classes — view the full character table.
| maximal quotient of | C3×AGL(1,𝔽7)×#997 |
Past the SmallGroups library coverage — no SmallGroups id.