Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| Fitting | C53×H(3,𝔽4) ⋊ C2 |
| Sylow | C53 ⋊ #3549 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | C53×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D53 |
| hypocenter | C53 | #3549 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 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=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^c=h^52,h^e=h^52532 conjugacy classes — view the full character table.
| maximal subgroup of | #27365903, #27365907, #27365937, … |
| maximal normal subgroup of | #27365903, #27365907, #27365937, … |
| maximal quotient of | #27546942, #27547608, #27547614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 18914);