Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #1830 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16811566 |
| commutator | C23×C53 | C22×C4 |
| Frattini | C23 | C22×#6157 |
| Fitting | #117×C53 | C4 |
| socle | C2×C53 | #381 |
| hypercenter | #117 | #6157 |
| hypocenter | C53 | #1830 |
| LCS | G ▷ C23×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 | |
236 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 62 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=d,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^30,f^b=fg,e^c=eg,h^f=h^52289 conjugacy classes — view the full character table.
| maximal subgroup of | #28311022, #28311023, #28311024, … |
| maximal normal subgroup of | #28311022, #28311023, #28311024, … |
| maximal quotient of | #28303389, #28303390, #28303391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20424);