Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2689 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator, socle | C23×C53 | C24 |
| Fitting | C2×#126×C53 | C2 |
| hypercenter | C2×#126 | D53 |
| hypocenter | C53 | #2689 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#126 | |
456 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=e,b^2=ef,c^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,h^c=h^52760 conjugacy classes — view the full character table.
| maximal subgroup of | #27590861, #27590870, #27590951, … |
| maximal normal subgroup of | #27590861, #27590870, #27590951, … |
| maximal quotient of | #27311468, #27311515, #27311710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 10955);