Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | #97×C53 ⋊ C4 |
| Sylow | C53 ⋊ #1833 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133296 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | #97×C53 | C4 |
| socle | C2×C53 | #382 |
| hypercenter | #97 | #6157 |
| hypocenter | C53 | #1833 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ #97 | |
172 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 62 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=e,c^2=d,e^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^30,f^b=fg,h^b=h^52,f^c=fg,h^f=h^52292 conjugacy classes — view the full character table.
| maximal subgroup of | #28311258, #28311259, #28311260, … |
| maximal normal subgroup of | #28311258, #28311259, #28311260, … |
| maximal quotient of | #28303519, #28303520, #28303523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20430);