Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C53 ⋊ M128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C2×D53 |
| commutator, socle | C2×C53 | C2×C32 |
| Fitting | C2×C32×C53 | C2 |
| hypercenter | C2×C32 | D53 |
| hypocenter | C53 | M128 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C32 ◁ C2×C32 | |
38 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,b^a=bg,h^a=h^521744 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27271898, #27271899, #27271902, … |
| maximal normal subgroup of | #27271898, #27271899, #27271902, … |
| maximal quotient of | #27271897, C22⋊#16810342, C4⋊#16810342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2721);