Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #2070 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810547 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#338 | C2 |
| socle | C2×C53 | #396 |
| hypercenter | #338 | D53 |
| hypocenter | C53 | #2070 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #338 | |
196 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=de,b^2=d,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^52,c^b=cf,f^b=fg,h^b=h^52,d^c=dg,h^c=h^52595 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2070, C2×#20276117, #27412815, … |
| maximal normal subgroup of | D53×#2070, C2×#20276117, #27412815, … |
| maximal quotient of | #27288729, #27288743, #27288757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5912);