Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#16811626 |
| Sylow | C53 ⋊ C2×#423 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C23×C4×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C23×C4 | #6157 |
| hypocenter | C53 | C2×#423 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
576 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
472 conjugacy classes — view the full character table.
| maximal subgroup of | #28300176, #28300182, #28300186, … |
| maximal normal subgroup of | #28300176, #28300182, #28300186, … |
| maximal quotient of | C2×#20290269, C2×#20290272, C2×#20290275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21145);