Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| Fitting | C53×#376 ⋊ C2 |
| Sylow | C53 ⋊ C4×#117 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D53 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C23 | C23×D53 |
| Fitting | C53×#376 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | #376 | D53 |
| hypocenter | C53 | C4×#117 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
752 conjugacy classes of subgroups; only the 341 normal subgroups are shown.
This group has 752 conjugacy classes of subgroups — too many to draw. Only its 341 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=f,b^2=g,c^2=f,d^2=e,b^a=be,h^c=h^521120 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27273489, #27273495, #27274286, … |
| maximal normal subgroup of | #27273489, #27273495, #27274286, … |
| maximal quotient of | #376×Dic53, C4×#16810470, C4×#16810474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6849);