Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#16810345 |
| Sylow | C53 ⋊ C2×#322 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D53 |
| commutator | C2×C53 | C2×C4×C8 |
| Frattini | C42 | C22×D53 |
| Fitting | C2×C4×C8×C53 | C2 |
| socle | C23×C53 | C42 |
| hypercenter | C2×C4×C8 | D53 |
| hypocenter | C53 | C2×#322 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C4×C8 | |
248 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1744 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27270940, #27270942, #27271066, … |
| maximal normal subgroup of | #27270940, #27270942, #27271066, … |
| maximal quotient of | C4×#16810345, C2×#20272542, #27274515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2786);