Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#518 ⋊ C2 |
| Sylow | C53 ⋊ #1913 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810502 |
| commutator | C23×C53 | C22×C4 |
| Frattini | C24 | C22×D53 |
| Fitting | C53×#518 | C2 |
| socle | C2×C53 | #386 |
| hypercenter | #518 | D53 |
| hypocenter | C53 | #1913 |
| LCS | G ▷ C23×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ #518 | |
380 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 89 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=f,b^2=df,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^52,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg913 conjugacy classes — view the full character table.
| maximal subgroup of | #27299849, #27299854, #27299856, … |
| maximal normal subgroup of | #27299849, #27299854, #27299856, … |
| maximal quotient of | #27279627, #27279632, #27279637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5052);