Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | #120×Dic53 |
| Sylow | C53 ⋊ C4×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C2×#120×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#120 | D53 |
| hypocenter | C53 | C4×#120 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
1224 conjugacy classes of subgroups; only the 289 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 289 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
784 conjugacy classes — view the full character table.
| maximal subgroup of | AΣL(1,𝔽16)×Dic53, #27300053, B(4,𝔽2)×Dic53, … |
| maximal normal subgroup of | AΣL(1,𝔽16)×Dic53, #27300053, B(4,𝔽2)×Dic53, … |
| maximal quotient of | #386×Dic53, #385×Dic53, #379×Dic53, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7013);