Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ C8×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×#6157 |
| commutator | C4×C53 | C22×C8 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | #107×C53 | C4 |
| socle | C22×C53 | C4×D4 |
| hypercenter | #107 | #6157 |
| hypocenter | C53 | C8×D8 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #107 | |
196 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 66 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=e,c^2=e,d^2=f,e^2=g,b^a=bd,d^a=df,h^a=h^23,d^b=df,h^b=h^52,h^c=h^30,h^e=h^52238 conjugacy classes — view the full character table.
| maximal subgroup of | #28309965, #28309966, #28309969, … |
| maximal normal subgroup of | #28309965, #28309966, #28309969, … |
| maximal quotient of | #28301748, #107×#88324, #28303671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20091);