Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#488 ⋊ C2 |
| Sylow | C53 ⋊ #3233 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#488 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #488 | D53 |
| hypocenter | C53 | #3233 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #488 | |
332 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|b^2=eg,c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,f^c=fg,h^d=h^52604 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#3233, C2×#20285889, #28256177, … |
| maximal normal subgroup of | D53×#3233, C2×#20285889, #28256177, … |
| maximal quotient of | #27393596, #27393791, #27394641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 15684);