Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C53×#533 ⋊ C2 |
| Sylow | C53 ⋊ #3455 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C53×#533 | C2 |
| hypercenter | #533 | D53 |
| hypocenter | C53 | #3455 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #533 | |
1040 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,h^a=h^52,c^b=cf,h^b=h^52,h^c=h^52772 conjugacy classes — view the full character table.
| maximal subgroup of | #27275938, #27275942, #27356397, … |
| maximal normal subgroup of | #27275938, #27275942, #27356397, … |
| maximal quotient of | #27542695, #27542755, #27542858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17902);