Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ Q8×C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×#6157 |
| commutator | C2×C53 | C22×C16 |
| Fitting | C4⋊C8×C53 | C4 |
| socle | C22×C53 | C22×C8 |
| hypercenter | C4⋊C8 | #6157 |
| hypocenter | C53 | Q8×C16 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
134 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=e,b^2=d,c^2=de,e^2=f,f^2=g,b^a=bd,h^a=h^23,h^c=h^30,h^e=h^52340 conjugacy classes — view the full character table.
| maximal subgroup of | #28309991, #28309993, #28309999, … |
| maximal normal subgroup of | #28309991, #28309993, #28309999, … |
| maximal quotient of | #28310774, #28310775, #28311845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20924);