Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#427 ⋊ C2 |
| Sylow | C53 ⋊ #2949 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C4×C53 | C23×C4 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#427 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #427 | D53 |
| hypocenter | C53 | #2949 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #427 | |
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,b^2=g,c^2=f,e^2=g,b^a=be,d^a=dg,e^a=eg,e^b=eg,h^b=h^52,d^c=dg,h^c=h^52772 conjugacy classes — view the full character table.
| maximal subgroup of | #27397993, #27397995, #27400003, … |
| maximal normal subgroup of | #27397993, #27397995, #27400003, … |
| maximal quotient of | #27353556, #27353771, #27353780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13368);