Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | D53⋊#446 |
| Sylow | C53 ⋊ C2×#446 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C4×Q8×C53 | C4 |
| hypercenter | C4×Q8 | #6157 |
| hypocenter | C53 | C2×#446 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
400 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=f,b^2=e,c^2=ef,f^2=g,b^a=be,c^a=cg,h^a=h^23,h^c=h^30,h^d=h^52,h^f=h^52316 conjugacy classes — view the full character table.
| maximal subgroup of | #28301785, #28301793, #28301801, … |
| maximal normal subgroup of | #28301785, #28301793, #28301801, … |
| maximal quotient of | #28318297, #28318302, C4×#16811673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21296);