Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #2166 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133296 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | #97×C53 | C4 |
| socle | C2×C53 | #432 |
| hypercenter | #97 | #6157 |
| hypocenter | C53 | #2166 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #97 | |
122 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,c^a=cf,d^a=dg,h^a=h^30,c^b=cg,e^b=eg,h^b=h^52,e^c=eg,h^c=h^52,h^e=h^52292 conjugacy classes — view the full character table.
| maximal subgroup of | #28298593, #28298596, #28309094, … |
| maximal normal subgroup of | #28298593, #28298596, #28309094, … |
| maximal quotient of | #28310447, #28310509, #28310770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20886);