Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #1728 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#6157 |
| commutator | C22×C53 | C2×C42 |
| Fitting | C23×C4×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C23×C4 | #6157 |
| hypocenter | C53 | #1728 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
480 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 158 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,c^a=cg,h^a=h^30,h^e=h^52472 conjugacy classes — view the full character table.
| maximal subgroup of | #28298047, #28298054, #28298077, … |
| maximal normal subgroup of | #28298047, #28298054, #28298077, … |
| maximal quotient of | #28300065, #28300075, #28300079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20173);