Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #2162 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D53 |
| commutator | C2×C53 | C22×C16 |
| Frattini | C2×C8 | C22×D53 |
| Fitting | C2×C4×C8×C53 | C2 |
| socle | C22×C53 | C22×C8 |
| hypercenter | C2×C4×C8 | D53 |
| hypocenter | C53 | #2162 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 | |
160 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,h^a=h^521744 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27270933, #27270934, #27270938, … |
| maximal normal subgroup of | #27270933, #27270934, #27270938, … |
| maximal quotient of | #27299091, C4×C22⋊#11719908, C4×C4⋊#11719908, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6309);