Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #2295 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C53×#389 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #389 | D53 |
| hypocenter | C53 | #2295 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #389 | |
500 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,b^2=fg,c^2=f,d^2=e,b^a=be,c^a=cf,h^d=h^52784 conjugacy classes — view the full character table.
| maximal subgroup of | #27272038, #27272098, #27272116, … |
| maximal normal subgroup of | #27272038, #27272098, #27272116, … |
| maximal quotient of | #27305198, #27305219, #27305272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6981);