Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×#13133277 |
| Sylow | C53 ⋊ C22×#100 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22⋊#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C25×C53 | C4 |
| socle | C23×C53 | C22⋊C4 |
| hypercenter | C25 | #6157 |
| hypocenter | C53 | C22×#100 |
| LCS | G ▷ C22×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C25 | |
992 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
460 conjugacy classes — view the full character table.
| maximal subgroup of | #28306591, #28306592, #28306613, … |
| maximal normal subgroup of | #28306591, #28306592, #28306613, … |
| maximal quotient of | C22×#16811466, C22×#16811467, C2×#20289954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21113);