Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#424 ⋊ C2 |
| Sylow | C53 ⋊ #2930 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | C53×#424 | C2 |
| hypercenter | #424 | D53 |
| hypocenter | C53 | #2930 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #424 | |
492 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 181 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=f,c^2=f,f^2=g,b^a=be,d^a=dg,c^b=cg,h^b=h^52,h^c=h^52772 conjugacy classes — view the full character table.
| maximal subgroup of | #27293397, #27389948, #27389952, … |
| maximal normal subgroup of | #27293397, #27389948, #27389952, … |
| maximal quotient of | #27354283, #27354292, #27354301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13212);