Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C22×#13133262 |
| Fitting | C23×D4×C53 ⋊ C2 |
| Sylow | C53 ⋊ C22×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C2×C53 | C26 |
| Frattini | C2 | C25×D53 |
| Fitting | C23×D4×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C23×D4 | D53 |
| hypocenter | C53 | C22×H(5,𝔽2) |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C23×D4 | |
6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.
This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1108 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27383225, #27383226, #27383228, … |
| maximal normal subgroup of | #27383225, #27383226, #27383228, … |
| maximal quotient of | C22×#16811095, C22×#16811096, C22×#16811098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19311);