Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #2098 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133213 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#421 | C2 |
| socle | C2×C53 | #399 |
| hypercenter | #421 | D53 |
| hypocenter | C53 | #2098 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #421 | |
210 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 65 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,d^2=g,f^2=g,b^a=bd,c^a=ce,f^a=fg,h^a=h^52,c^b=cf,d^b=dg,e^b=eg,f^b=fg,f^c=fg598 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2098, C2×#20276270, #27413978, … |
| maximal normal subgroup of | D53×#2098, C2×#20276270, #27413978, … |
| maximal quotient of | #27289061, #27289065, #27289068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6065);