Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×2−1+4×C53 ⋊ C2 |
| Sylow | C53 ⋊ #3487 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | C2×2−1+4×C53 | C2 |
| hypercenter | C2×2−1+4 | D53 |
| hypocenter | C53 | #3487 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
1536 conjugacy classes of subgroups; only the 549 normal subgroups are shown.
This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 549 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=f,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^c=h^52928 conjugacy classes — view the full character table.
| maximal subgroup of | #27299652, #27393270, #27415520, … |
| maximal normal subgroup of | #27299652, #27393270, #27415520, … |
| maximal quotient of | #27544120, #27544382, #27545027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 18165);