Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#471 ⋊ C2 |
| Sylow | C53 ⋊ #3069 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #121×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#471 | C2 |
| socle | C2×C53 | C2×#121 |
| hypercenter | #471 | D53 |
| hypocenter | C53 | #3069 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #471 | |
446 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,b^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg,f^c=fg,h^d=h^52445 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#3069, C2×#20284447, #28251510, … |
| maximal normal subgroup of | D53×#3069, C2×#20284447, #28251510, … |
| maximal quotient of | #27366864, #27366866, #27366875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 14242);