Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #2108 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D53 |
| commutator | C2×C53 | C2×C4×C8 |
| Fitting | C2×M32×C53 | C2 |
| socle | C22×C53 | C22×C8 |
| hypercenter | C2×M32 | D53 |
| hypocenter | C53 | #2108 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C2×M32 | |
184 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 125 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=d,b^2=e,d^2=f,f^2=g,h^a=h^52,c^b=cg,h^b=h^521120 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27298926, #27299113, #27299187, … |
| maximal normal subgroup of | #27298926, #27299113, #27299187, … |
| maximal quotient of | #27293076, #27293083, #27293107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6109);