Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C53 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810458 |
| commutator | C32×C53 | C22 |
| Frattini | C32 | C2×D53 |
| Fitting | C53×Q64 | C2 |
| socle | C2×C53 | D32 |
| hypercenter | Q64 | D53 |
| hypocenter | C53 | Q128 |
| LCS | G ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^52529 conjugacy classes — view the full character table.
| maximal subgroup of | D53×Q128, C2×#20272932, #27305144, … |
| maximal normal subgroup of | D53×Q128, C2×#20272932, #27305144, … |
| maximal quotient of | #27271972, #27271973, #27271991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2727);