Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #2102 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133206 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#423 | C2 |
| socle | C2×C53 | #400 |
| hypercenter | #423 | D53 |
| hypocenter | C53 | #2102 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #423 | |
158 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 158 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=d,b^2=de,c^2=d,d^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,h^b=h^52,f^c=fg754 conjugacy classes — view the full character table.
| maximal subgroup of | #27271712, #27271714, #27271715, … |
| maximal normal subgroup of | #27271712, #27271714, #27271715, … |
| maximal quotient of | #27289217, #27289222, #27289227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6085);