Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C22×C33 |
| direct product | C2×A42×ΓL(1,𝔽8) |
| Fitting, socle, Hall | C25×C7 ⋊ C33 |
| derived | C24×C7 ⋊ C2×C33 |
| Sylow | C25 ⋊ C32×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C2×C3×A42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A42×ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C24×C7 | C2×C33 |
| Frattini | C1 | G |
| Fitting, socle | C25×C7 | C33 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C2 | |
824 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 824 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 126 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A42×AGL(1,𝔽7), C2×ΓL(1,𝔽8)×A4⋊S4, C2×#72222243 |
| maximal normal subgroup of | C2×A42×AGL(1,𝔽7), C2×ΓL(1,𝔽8)×A4⋊S4, C2×#72222243 |
Past the SmallGroups library coverage — no SmallGroups id.