Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C2×C33 |
| direct product | ΓL(1,𝔽8)×Ω+(4,𝔽3) |
| Fitting, derived, Hall | C7×H(5,𝔽2) ⋊ C33 |
| Sylow | H(5,𝔽2) ⋊ C32×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×Ω+(4,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A42×ΓL(1,𝔽8) |
| commutator, Fitting, cosocle, hypocenter | C7×H(5,𝔽2) | C33 |
| socle | C2×C7 | C3×A42 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
400 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
125 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×Ω+(4,𝔽3), ΓL(1,𝔽8)×SO+(4,𝔽3), #72246260 |
| maximal normal subgroup of | AGL(1,𝔽7)×Ω+(4,𝔽3), ΓL(1,𝔽8)×SO+(4,𝔽3), #72246260 |
Past the SmallGroups library coverage — no SmallGroups id.