Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #2110 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C4×Dic53 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C2×C8 | C22×D53 |
| Fitting | C53×#405 | C2 |
| socle | C2×C53 | C2×#97 |
| hypercenter | #405 | D53 |
| hypocenter | C53 | #2110 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ #405 | |
160 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=d,b^2=e,d^2=f,f^2=g,b^a=bf,e^a=eg,h^a=h^52,c^b=cg,d^b=dg1084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27295274, #27295275, #27295278, … |
| maximal normal subgroup of | #27295274, #27295275, #27295278, … |
| maximal quotient of | #27293097, #27293101, #27293134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6115);