Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C53×#407 ⋊ C2 |
| Sylow | C53 ⋊ #3004 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#407 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #407 | D53 |
| hypocenter | C53 | #3004 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #407 | |
738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 738 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|b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=h^52,e^b=eg,h^b=h^52760 conjugacy classes — view the full character table.
| maximal subgroup of | #27282110, #27282112, #27356496, … |
| maximal normal subgroup of | #27282110, #27282112, #27356496, … |
| maximal quotient of | #27356472, #27356511, #27356550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13747);