Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C53×#491 ⋊ C2 |
| Sylow | C53 ⋊ #3223 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#491 | C2 |
| socle | C22×C53 | C22×D4 |
| hypercenter | #491 | D53 |
| hypocenter | C53 | #3223 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #491 | |
364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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=g,b^2=e,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^52,c^b=ce,d^b=dg,f^c=fg,h^d=h^52442 conjugacy classes — view the full character table.
| maximal subgroup of | #27304512, D53×#3223, C2×#20285799, … |
| maximal normal subgroup of | #27304512, D53×#3223, C2×#20285799, … |
| maximal quotient of | #27393478, #27393892, #27394695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 15594);