Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #2108 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×#6157 |
| commutator | C2×C53 | C2×C4×C8 |
| Fitting | C2×M16×C53 | C4 |
| socle | C22×C53 | C22×C8 |
| hypercenter | C2×M16 | #6157 |
| hypocenter | C53 | #2108 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×M16 | |
184 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 114 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=d,b^2=e,d^2=f,f^2=g,h^a=h^30,c^b=cg,h^b=h^52,h^d=h^52340 conjugacy classes — view the full character table.
| maximal subgroup of | #28310208, #28310209, #28310453, … |
| maximal normal subgroup of | #28310208, #28310209, #28310453, … |
| maximal quotient of | #28307757, #28307766, M16×#11719910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20721);