Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2960 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C53×#533 | C2 |
| hypercenter | #533 | D53 |
| hypocenter | C53 | #2960 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #533 | |
364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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=e,c^2=ef,f^2=g,b^a=be,d^a=dg,h^a=h^52,h^c=h^52784 conjugacy classes — view the full character table.
| maximal subgroup of | #27275102, #27275103, #27275112, … |
| maximal normal subgroup of | #27275102, #27275103, #27275112, … |
| maximal quotient of | #27382044, #27382061, #27382709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13445);