Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ C2×#336 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×#6157 |
| commutator | C2×C53 | C2×C4×C8 |
| Frattini | C23 | C22×#6157 |
| Fitting | C2×C42×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C42 | #6157 |
| hypocenter | C53 | C2×#336 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 | |
528 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 206 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,h^c=h^52,h^e=h^52496 conjugacy classes — view the full character table.
| maximal subgroup of | #28298009, #28298024, #28298033, … |
| maximal normal subgroup of | #28298009, #28298024, #28298033, … |
| maximal quotient of | D53⋊#1273, #28300064, #28300070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20171);