Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #2381 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C2×C4⋊Q8×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×C4⋊Q8 | D53 |
| hypocenter | C53 | #2381 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
424 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=def,h^a=h^52,c^b=cef,d^b=de766 conjugacy classes — view the full character table.
| maximal subgroup of | #27281632, #27281635, #27281637, … |
| maximal normal subgroup of | #27281632, #27281635, #27281637, … |
| maximal quotient of | #27305719, #27305742, #27305746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7691);