Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#349 ⋊ C2 |
| Sylow | C53 ⋊ #2115 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊C4×D53 |
| commutator | C22×C53 | C22×C8 |
| Frattini | C2×C8 | C22×D53 |
| Fitting | C53×#349 | C2 |
| socle | C2×C53 | C2×C22⋊C8 |
| hypercenter | #349 | D53 |
| hypocenter | C53 | #2115 |
| LCS | G ▷ C22×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #349 | |
220 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 75 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,e^2=f,f^2=g,b^a=bd,d^a=dg,c^b=cg,e^b=eg,h^c=h^52616 conjugacy classes — view the full character table.
| maximal subgroup of | #27293391, #27293393, #27293395, … |
| maximal normal subgroup of | #27293391, #27293393, #27293395, … |
| maximal quotient of | #27293155, #27293227, #27293295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6129);