Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C4×#41×C53 ⋊ C2 |
| Sylow | C53 ⋊ #3460 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C4×#41×C53 | C2 |
| hypercenter | C4×#41 | D53 |
| hypocenter | C53 | #3460 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 | |
1140 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|b^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=cf,h^c=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27275499, #27275502, #27350666, … |
| maximal normal subgroup of | #27275499, #27275502, #27350666, … |
| maximal quotient of | #27542775, #27542791, #27542799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17926);