Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C53×#576 ⋊ C2 |
| Sylow | C53 ⋊ #3584 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D4×D53 |
| commutator | C4×C53 | C25 |
| Frattini | C4 | C24×D53 |
| Fitting | C53×#576 | C2 |
| socle | C2×C53 | C23×D4 |
| hypercenter | #576 | D53 |
| hypocenter | C53 | #3584 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #576 | |
1424 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|e^2=g,f^2=g,b^a=bf,f^a=fg,f^b=fg,d^c=dg,h^e=h^52616 conjugacy classes — view the full character table.
| maximal subgroup of | #27538043, #27538049, #27538072, … |
| maximal normal subgroup of | #27538043, #27538049, #27538072, … |
| maximal quotient of | #28247652, #28247769, #28248205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19278);