Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#16811352 |
| Sylow | C53 ⋊ C2×#557 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×C4×Q8×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×C4×Q8 | D53 |
| hypocenter | C53 | C2×#557 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
1048 conjugacy classes of subgroups; only the 581 normal subgroups are shown.
This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 581 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1090 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27275829, #27275838, #27275841, … |
| maximal normal subgroup of | #27275829, #27275838, #27275841, … |
| maximal quotient of | C2×Q8×C4⋊Dic53, C2×#20277633, C2×#20277670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 18083);