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 ⋊ C8×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×#6157 |
| commutator | C2×C53 | C2×C4×C8 |
| Fitting | #97×C53 | C4 |
| socle | C22×C53 | C2×C42 |
| hypercenter | #97 | #6157 |
| hypocenter | C53 | C8×M16 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #97 | |
224 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 110 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,e^2=g,c^a=cf,h^b=h^30,h^c=h^52,h^e=h^52340 conjugacy classes — view the full character table.
| maximal subgroup of | #28301929, #28301933, #28301947, … |
| maximal normal subgroup of | #28301929, #28301933, #28301947, … |
| maximal quotient of | #97×#88324, #28300064, #28301456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19708);