Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #1729 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#6157 |
| commutator | C22×C53 | C2×C42 |
| Fitting | C2×C22⋊C4×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C22⋊C4 | #6157 |
| hypocenter | C53 | #1729 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
416 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 146 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=e,b^2=f,e^2=g,b^a=bd,h^a=h^30,c^b=cg,h^e=h^52316 conjugacy classes — view the full character table.
| maximal subgroup of | #28298420, #28306540, #28306544, … |
| maximal normal subgroup of | #28298420, #28306540, #28306544, … |
| maximal quotient of | #28300067, #28300071, #28300073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20176);