Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #1746 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D53 |
| commutator | C22×C53 | C2×C42 |
| Fitting | C2×C4×C8×C53 | C2 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C4×C8 | D53 |
| hypocenter | C53 | #1746 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 | |
240 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=e,b^2=f,c^2=d,e^2=g,b^a=bd,c^b=cg,h^b=h^521720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27270937, #27270939, #27274547, … |
| maximal normal subgroup of | #27270937, #27270939, #27274547, … |
| maximal quotient of | #27274518, #27274521, #27274524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4261);