Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×#131×C53 ⋊ C2 |
| Sylow | C53 ⋊ #3584 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D212 |
| commutator | C4×C53 | C25 |
| Frattini | C4 | C24×D53 |
| Fitting | C2×#131×C53 | C2 |
| socle | C2×C53 | C23×D4 |
| hypercenter | C2×#131 | D53 |
| hypocenter | C53 | #3584 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ C2×#131 | |
1424 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|e^2=g,f^2=g,b^a=bf,f^a=fg,h^a=h^52,f^b=fg,h^b=h^52,d^c=dg1084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27299797, #27299800, #27299802, … |
| maximal normal subgroup of | #27299797, #27299800, #27299802, … |
| maximal quotient of | #28247658, #28247771, #28248213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19284);