Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | D53⋊#424 |
| Sylow | C53 ⋊ C2×#424 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | #117×C53 | C4 |
| hypercenter | #117 | #6157 |
| hypocenter | C53 | C2×#424 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
512 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 210 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=e,f^2=g,b^a=be,h^a=h^30,c^b=cg,h^d=h^52,h^f=h^52316 conjugacy classes — view the full character table.
| maximal subgroup of | #28306948, #28306951, #28306963, … |
| maximal normal subgroup of | #28306948, #28306951, #28306963, … |
| maximal quotient of | D53⋊#1570, D53⋊#1571, D53⋊#1572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21150);