Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C53×#553 ⋊ C2 |
| Sylow | C53 ⋊ #3551 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | C53×#553 | C2 |
| hypercenter | #553 | D53 |
| hypocenter | C53 | #3551 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #553 | |
1050 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^b=h^52,h^c=h^52,h^d=h^52688 conjugacy classes — view the full character table.
| maximal subgroup of | #28270846, #28270850, #28270887, … |
| maximal normal subgroup of | #28270846, #28270850, #28270887, … |
| maximal quotient of | #27546948, #27547315, #27547318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 18936);