Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C210 |
| direct product | C23×C3×C4×D7 |
| Fitting | C23×C3×C4×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C24×C3×C4 |
| Hall | C3×C7 ⋊ C24×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3×C4 | D7 |
| commutator, hypocenter | C7 | C24×C3×C4 |
| Frattini | C2 | C24×C3×D7 |
| Fitting | C23×C3×C4×C7 | C2 |
| socle | C24×C3×C7 | C22 |
| prime radical | C24×C3×D7 | C2 |
| cosocle | C2×C7 | C25×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C23×C3×C4 | |
2724 conjugacy classes of subgroups; only the 1598 normal subgroups are shown.
This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 1598 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 672 |
480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11618);