Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C53×#433 ⋊ C2 |
| Sylow | C53 ⋊ #3222 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#433 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #433 | D53 |
| hypocenter | C53 | #3222 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #433 | |
396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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=g,b^2=e,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^52,c^b=ce,d^b=dg,f^c=fg,h^c=h^52754 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#3222, C2×#20285789, #28255498, … |
| maximal normal subgroup of | D53×#3222, C2×#20285789, #28255498, … |
| maximal quotient of | #27393468, #27393883, #27394681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 15584);