Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C53×#347 ⋊ C2 |
| Sylow | C53 ⋊ #2165 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133174 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C2×C8 | C22×D53 |
| Fitting | C53×#347 | C2 |
| socle | C2×C53 | #432 |
| hypercenter | #347 | D53 |
| hypocenter | C53 | #2165 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ #97 ◁ #347 | |
128 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 53 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^a=cf,d^a=dg,e^b=eg,h^b=h^52,e^c=eg604 conjugacy classes — view the full character table.
| maximal subgroup of | #27295284, #27295287, #27295290, … |
| maximal normal subgroup of | #27295284, #27295287, #27295290, … |
| maximal quotient of | #27299106, #27299159, #27299427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6320);