Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#338 ⋊ C2 |
| Sylow | C53 ⋊ #2091 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133214 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#338 | C2 |
| socle | C2×C53 | #399 |
| hypercenter | #338 | D53 |
| hypocenter | C53 | #2091 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #338 | |
222 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=d,b^2=de,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,h^c=h^52598 conjugacy classes — view the full character table.
| maximal subgroup of | #27297418, #27297423, #27297429, … |
| maximal normal subgroup of | #27297418, #27297423, #27297429, … |
| maximal quotient of | #27289062, #27289069, #27289074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6027);