Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C53⋊#2104 |
| Sylow | C53 ⋊ #2104 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C53⋊#401 |
| commutator | C24×C53 | C23 |
| Frattini | C24 | C22×D53 |
| Fitting | C53×#342 | C2 |
| socle | C2×C53 | #401 |
| hypercenter | #342 | D53 |
| hypocenter | C53 | #2104 |
| LCS | G ▷ C24×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ #342 | |
178 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 53 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=def,b^2=de,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,f^b=fg,e^c=eg,h^c=h^52595 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2104, C2×C53⋊#2104, #27415081, … |
| maximal normal subgroup of | D53×#2104, C2×C53⋊#2104, #27415081, … |
| maximal quotient of | #27289233, #27289234, #27289235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6094);