Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #1752 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D53 |
| commutator | C22×C53 | C2×C42 |
| Fitting | C23×C8×C53 | C2 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C23×C8 | D53 |
| hypocenter | C53 | #1752 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 | |
360 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=e,b^2=f,c^2=e,e^2=g,b^a=bd,c^b=cg,h^b=h^521720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27271077, #27271078, #27271080, … |
| maximal normal subgroup of | #27271077, #27271078, #27271080, … |
| maximal quotient of | #27274561, #27274565, #27274587, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4286);