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 ⋊ #2337 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C2×#124×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#124 | D53 |
| hypocenter | C53 | #2337 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#124 | |
540 conjugacy classes of subgroups; only the 217 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 217 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=eg,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=de,h^a=h^52,c^b=ce,h^b=h^52772 conjugacy classes — view the full character table.
| maximal subgroup of | #27281377, #27281381, #27281391, … |
| maximal normal subgroup of | #27281377, #27281381, #27281391, … |
| maximal quotient of | #27305366, #27305423, #27305443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7325);