Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×Q8×#6157 |
| Fitting | C4×Q8×C53 ⋊ C4 |
| Sylow | C53 ⋊ C42×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#6157 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C22 | C23×#6157 |
| Fitting | C4×Q8×C53 | C4 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C4×Q8 | #6157 |
| hypocenter | C53 | C42×Q8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
684 conjugacy classes of subgroups; only the 320 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 320 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
340 conjugacy classes — view the full character table.
| maximal subgroup of | Q8⋊C8×#6157, #28301633, #28301639, … |
| maximal normal subgroup of | Q8⋊C8×#6157, #28301633, #28301639, … |
| maximal quotient of | C4×C4⋊C4×#6157, C4×#16811572, #387×#6157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20945);