Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C23×C33 |
| direct product | C3×ΓL(1,𝔽8)×#997 |
| Fitting | C3×C7×H(5,𝔽2) ⋊ C32 |
| derived, Hall | C7×H(5,𝔽2) ⋊ C33 |
| Sylow | H(5,𝔽2) ⋊ C32×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C32×#997 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | ΓL(1,𝔽8)×C22⋊A4 |
| commutator, cosocle, hypocenter | C7×H(5,𝔽2) | C33 |
| Frattini | C2 | C3×ΓL(1,𝔽8)×C22⋊A4 |
| Fitting | C3×C7×H(5,𝔽2) | C32 |
| socle | C2×C3×C7 | C3×C22⋊A4 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
904 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 904 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
165 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(1,𝔽7)×#997, D3×ΓL(1,𝔽8)×#997, #997×C3⋊AGL(1,𝔽7), … |
| maximal normal subgroup of | C3×AGL(1,𝔽7)×#997, D3×ΓL(1,𝔽8)×#997, #997×C3⋊AGL(1,𝔽7), … |
Past the SmallGroups library coverage — no SmallGroups id.