Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C53×#568 ⋊ C2 |
| Sylow | C53 ⋊ #3586 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×D212 |
| commutator | C4×C53 | C25 |
| Frattini | C4 | C24×D53 |
| Fitting | C53×#568 | C2 |
| socle | C2×C53 | C23×D4 |
| hypercenter | #568 | D53 |
| hypocenter | C53 | #3586 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #568 | |
1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|b^2=g,f^2=g,b^a=bf,e^a=eg,f^a=fg,h^a=h^52,f^b=fg,h^b=h^52,d^c=dg925 conjugacy classes — view the full character table.
| maximal subgroup of | #27415399, #27415400, #27415421, … |
| maximal normal subgroup of | #27415399, #27415400, #27415421, … |
| maximal quotient of | #28247670, #28247677, #28247777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19298);