Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #2030 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133187 |
| commutator | C24×C53 | C23 |
| Frattini | C24 | C22×D53 |
| Fitting | C53×#342 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | #342 | D53 |
| hypocenter | C53 | #2030 |
| LCS | G ▷ C24×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #342 | |
290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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=d,b^2=d,c^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,e^b=eg,h^b=h^52,h^c=h^52598 conjugacy classes — view the full character table.
| maximal subgroup of | #27296821, #27296823, #27296834, … |
| maximal normal subgroup of | #27296821, #27296823, #27296834, … |
| maximal quotient of | #27287071, #27287073, #27287076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5678);