Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #97×C53 ⋊ C4 |
| Sylow | C53 ⋊ #1841 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133296 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | #97×C53 | C4 |
| socle | C2×C53 | #383 |
| hypercenter | #97 | #6157 |
| hypocenter | C53 | #1841 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #97 | |
162 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 52 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=f,b^2=de,c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^30,h^b=h^52,f^c=fg,h^f=h^52292 conjugacy classes — view the full character table.
| maximal subgroup of | #28298553, #28298558, #28309327, … |
| maximal normal subgroup of | #28298553, #28298558, #28309327, … |
| maximal quotient of | #28303561, #28303562, #28303565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20441);