Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C4×C16×C53 ⋊ C2 |
| Sylow | C53 ⋊ #2164 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D53 |
| commutator, socle | C22×C53 | C22×C8 |
| Fitting | C4×C16×C53 | C2 |
| hypercenter | C4×C16 | D53 |
| hypocenter | C53 | #2164 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 | |
138 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 69 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,c^2=d,e^2=f,f^2=g,b^a=bd,c^b=cg,h^b=h^521720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×#2164, C2×#20276520, #27537175, … |
| maximal normal subgroup of | D53×#2164, C2×#20276520, #27537175, … |
| maximal quotient of | #27299096, #27299128, #27299406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6315);