Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C4 |
| direct product | AGL(1,𝔽7)×C42⋊C3 |
| Fitting, derived | C42×C7 ⋊ C2×C32 |
| Hall | C42×D7 ⋊ C32 |
| Sylow | C7 ⋊ C2×C3×C42⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C42×C7 | C2×C32 |
| Frattini | C22 | A4×AGL(1,𝔽7) |
| socle | C22×C7 | C2×C3×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
136 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#5464, AGL(1,𝔽7)×#127⋊C3, AGL(1,𝔽7)×#126⋊C3 |
| maximal normal subgroup of | AGL(1,𝔽7)×#127⋊C3, AGL(1,𝔽7)×#126⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id.