Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | #120×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C3×#120) |
| Fitting | C7×#120 ⋊ C2×C3 |
| Hall | D7×#120 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×#120 |
| Sylow | C7 ⋊ C2×C3×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#120 | C2×C3 |
| hypercenter | #120 | AGL(1,𝔽7) |
| hypocenter | C7 | C2×C3×#120 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #120 | |
1324 conjugacy classes of subgroups; only the 236 normal subgroups are shown.
This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 236 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6612);