Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #1375 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×#6157 |
| commutator | C4×C53 | C4×C8 |
| Frattini | C22×C4 | C2×#6157 |
| Fitting | C22×C8×C53 | C4 |
| socle | C22×C53 | #97 |
| hypercenter | C22×C8 | #6157 |
| hypocenter | C53 | #1375 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 | |
140 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=d,b^2=e,c^2=g,d^2=f,f^2=g,b^a=bc,e^a=eg,h^a=h^30,d^b=dg,h^b=h^52,h^d=h^52460 conjugacy classes — view the full character table.
| maximal subgroup of | #28298199, #28298200, #28298231, … |
| maximal normal subgroup of | #28298199, #28298200, #28298231, … |
| maximal quotient of | #28298258, #28298271, #28298280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19552);