Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #3390 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#499 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #499 | D53 |
| hypocenter | C53 | #3390 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #499 | |
320 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 121 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=e,c^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^52,c^b=cg,d^b=de,h^b=h^52,f^c=fg,h^c=h^52601 conjugacy classes — view the full character table.
| maximal subgroup of | #27297828, #27297832, #27297842, … |
| maximal normal subgroup of | #27297828, #27297832, #27297842, … |
| maximal quotient of | #27400784, #27402416, #27402471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17480);