Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×#117×C53 ⋊ C2 |
| Sylow | C53 ⋊ #3428 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D53 |
| commutator | C2×C53 | C24×C4 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×#117×C53 | C2 |
| socle | C22×C53 | C25 |
| hypercenter | C2×#117 | D53 |
| hypocenter | C53 | #3428 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 | |
1540 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,e^2=f,b^a=bf,d^a=df,c^b=cf,h^c=h^521108 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27289423, #27289426, #27289443, … |
| maximal normal subgroup of | #27289423, #27289426, #27289443, … |
| maximal quotient of | #27542304, #27542306, #27542318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17711);