Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C8×C22⋊#6157 |
| Fitting | C22×C8×C53 ⋊ C4 |
| Sylow | C53 ⋊ C8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×#6157 |
| commutator | C2×C53 | C2×C4×C8 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | C22×C8×C53 | C4 |
| socle | C22×C53 | C2×C42 |
| hypercenter | C22×C8 | #6157 |
| hypocenter | C53 | C8×C22⋊C4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 | |
384 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
496 conjugacy classes — view the full character table.
| maximal subgroup of | #28298451, #28298452, #28298463, … |
| maximal normal subgroup of | #28298451, #28298452, #28298463, … |
| maximal quotient of | C8×C22⋊#88324, C8×#13133274, #28301496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20242);