Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×C4×C3⋊(C7⋊C8) |
| derived, Hall | C3×C7 ⋊ C2×C4×C8 |
| Sylow | C7 ⋊ C2×C4×C3⋊C8 |
| Sylow | C3 ⋊ C2×C4×C7⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | D21 |
| commutator, hypocenter | C3×C7 | C2×C4×C8 |
| Frattini | C2×C4 | C22×D21 |
| Fitting | C2×C3×C42×C7 | C2 |
| socle, prime radical | C23×C3×C7 | C2×C4 |
| cosocle | C2×C3×C4×C7 | C23 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C42 | |
324 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1344 |
384 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5846);