Finite Group DatabasebetaFinite Group Databasebeta| order | 12348 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | AGL(1,𝔽7)×#64423 |
| Fitting, socle, derived, Sylow | C73 ⋊ C22×C32 |
| Hall | ΓL(1,𝔽8)×#3861 ⋊ C22 |
| Hall | C7×D72 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C73 | C22×C32 |
| LCS | G ▷ C73 | |
| UCS | C1 | |
340 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 340 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) = 84 |
119 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#64423⋊C2 |
| maximal normal subgroup of | AGL(1,𝔽7)×#64423⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id.