Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×#117×C53 ⋊ C2 |
| Sylow | C53 ⋊ #2392 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D53 |
| commutator, socle | C23×C53 | C24 |
| Frattini | C23 | C23×D53 |
| Fitting | C2×#117×C53 | C2 |
| hypercenter | C2×#117 | D53 |
| hypocenter | C53 | #2392 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#117 | |
696 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,h^c=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27279189, #27279191, #27279194, … |
| maximal normal subgroup of | #27279189, #27279191, #27279194, … |
| maximal quotient of | #27305768, #27305787, #27305828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7746);