Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ D4×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C4×D4×C53 | C4 |
| hypercenter | C4×D4 | #6157 |
| hypocenter | C53 | D4×M16 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
544 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,h^a=h^23,h^b=h^52,d^c=dg,h^c=h^30,h^d=h^52,h^f=h^52310 conjugacy classes — view the full character table.
| maximal subgroup of | #28303813, #28303815, #28303821, … |
| maximal normal subgroup of | #28303813, #28303815, #28303821, … |
| maximal quotient of | #28317493, #28317499, #28317507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21251);