Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×#13133182 |
| Sylow | C53 ⋊ C22×C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C2×C53 | C24×C4 |
| Frattini | C22 | C24×D53 |
| Fitting | C22×C4⋊C4×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C22×C4⋊C4 | D53 |
| hypocenter | C53 | C22×C4×Q8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C22×C4⋊C4 | |
1664 conjugacy classes of subgroups; only the 941 normal subgroups are shown.
This group has 1664 conjugacy classes of subgroups — too many to draw. Only its 941 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1120 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27275434, #27289519, #27289660, … |
| maximal normal subgroup of | #27275434, #27289519, #27289660, … |
| maximal quotient of | C22×#16810474, C22×C4⋊C4×Dic53, C2×C4×#13133182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17684);