Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#473 ⋊ C2 |
| Sylow | C53 ⋊ #3067 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #121×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#473 | C2 |
| socle | C2×C53 | C2×#121 |
| hypercenter | #473 | D53 |
| hypocenter | C53 | #3067 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #473 | |
452 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|b^2=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^52,h^b=h^52,f^c=fg,h^d=h^52448 conjugacy classes — view the full character table.
| maximal subgroup of | #27538049, #27538051, #27538059, … |
| maximal normal subgroup of | #27538049, #27538051, #27538059, … |
| maximal quotient of | #27366336, #27366360, #27366403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 14232);