Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#417 ⋊ C2 |
| Sylow | C53 ⋊ #2952 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#417 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #417 | D53 |
| hypocenter | C53 | #2952 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #417 | |
484 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,b^a=be,d^a=dg,e^a=eg,e^b=eg,d^c=dg,h^c=h^52616 conjugacy classes — view the full character table.
| maximal subgroup of | #27400067, #27400072, #27400093, … |
| maximal normal subgroup of | #27400067, #27400072, #27400093, … |
| maximal quotient of | #27353577, #27353806, #27353815, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13389);