Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #2080 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133204 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#421 | C2 |
| socle | C2×C53 | #397 |
| hypercenter | #421 | D53 |
| hypocenter | C53 | #2080 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #421 | |
190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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,d^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^c=fg,h^c=h^52598 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2080, C2×#20276165, #27413017, … |
| maximal normal subgroup of | D53×#2080, C2×#20276165, #27413017, … |
| maximal quotient of | #27288860, #27288865, #27288870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5960);