Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×#13133305 |
| Sylow | C53 ⋊ C22×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C2×C53 | C24×C4 |
| Frattini | C2 | C24×#6157 |
| Fitting | C22×Q8×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C22×Q8 | #6157 |
| hypocenter | C53 | C22×#131 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
1504 conjugacy classes of subgroups; only the 770 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 770 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 | #28306600, #28306676, #28317933, … |
| maximal normal subgroup of | #28306600, #28306676, #28317933, … |
| maximal quotient of | C22×#16811588, C2×C4×#13133305, C22×#16811598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21500);