Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#494 ⋊ C2 |
| Sylow | C53 ⋊ #1689 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133188 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×#494 | C2 |
| socle | C22×C53 | #121 |
| hypercenter | #494 | D53 |
| hypocenter | C53 | #1689 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #494 | |
166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf,e^c=eg,h^c=h^52595 conjugacy classes — view the full character table.
| maximal subgroup of | #27297583, #27297585, #27297590, … |
| maximal normal subgroup of | #27297583, #27297585, #27297590, … |
| maximal quotient of | #27277183, #27280318, #27282458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 3996);