Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#342 ⋊ C2 |
| Sylow | C53 ⋊ #1819 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22⋊C4×D53 |
| commutator | C23×C53 | C22×C4 |
| Frattini | C24 | C22×D53 |
| Fitting | C53×#342 | C2 |
| socle | C22×C53 | C2×C22⋊C4 |
| hypercenter | #342 | D53 |
| hypocenter | C53 | #1819 |
| LCS | G ▷ C23×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #342 | |
504 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 81 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,c^2=d,b^a=bd,c^a=ce,e^a=eg,h^b=h^52,f^c=fg604 conjugacy classes — view the full character table.
| maximal subgroup of | #27294746, #27294750, #27294759, … |
| maximal normal subgroup of | #27294746, #27294750, #27294759, … |
| maximal quotient of | #27277414, #27277420, #27277426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4583);