Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C53 ⋊ C4×#124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C53×#390 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #390 | D53 |
| hypocenter | C53 | C4×#124 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #390 | |
652 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=g,b^2=f,c^2=e,d^2=g,b^a=be,c^a=cf,h^a=h^52,h^b=h^52,h^c=h^52,h^d=h^52784 conjugacy classes — view the full character table.
| maximal subgroup of | #27272058, #27272131, #27272204, … |
| maximal normal subgroup of | #27272058, #27272131, #27272204, … |
| maximal quotient of | #27305253, #390×Dic53, #27305290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7057);