Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C53×#543 ⋊ C2 |
| Sylow | C53 ⋊ #3459 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | C53×#543 | C2 |
| hypercenter | #543 | D53 |
| hypocenter | C53 | #3459 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #543 | |
1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|e^2=g,b^a=bf,c^a=cg,d^a=df,h^a=h^52,c^b=cf,h^c=h^52,h^e=h^52616 conjugacy classes — view the full character table.
| maximal subgroup of | #27356197, #27356203, #27356248, … |
| maximal normal subgroup of | #27356197, #27356203, #27356248, … |
| maximal quotient of | #27542766, #27543039, #27543054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17922);