Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2987 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C53×#533 | C2 |
| hypercenter | #533 | D53 |
| hypocenter | C53 | #2987 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #533 | |
314 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 197 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=e,f^2=g,b^a=be,c^a=cg,d^a=de,h^a=h^52,c^b=ceg,d^b=dg766 conjugacy classes — view the full character table.
| maximal subgroup of | #27278499, #27278502, #27279044, … |
| maximal normal subgroup of | #27278499, #27278502, #27279044, … |
| maximal quotient of | #27382856, #27382877, #27383192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13616);