Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#13133123 |
| Sylow | C53 ⋊ C4⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C16 | C2×D53 |
| commutator | C2×C53 | C2×C32 |
| Fitting | C4×C16×C53 | C2 |
| socle | C22×C53 | C2×C16 |
| hypercenter | C4×C16 | D53 |
| hypocenter | C53 | C4⋊C32 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C16 | |
62 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=d,b^2=c,d^2=e,e^2=f,f^2=g,b^a=bc,h^a=h^521744 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D4⋊#13133123, #27271531, #27271533, … |
| maximal normal subgroup of | D4⋊#13133123, #27271531, #27271533, … |
| maximal quotient of | #27271758, #27271765, #27271779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2706);