Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | #124×#6157 |
| Fitting | #124×C53 ⋊ C4 |
| Sylow | C53 ⋊ C4×#124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #124×C53 | C4 |
| hypercenter | #124 | #6157 |
| hypocenter | C53 | C4×#124 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #124 | |
652 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
238 conjugacy classes — view the full character table.
| maximal subgroup of | #329×#6157, #28301862, #28301870, … |
| maximal normal subgroup of | #329×#6157, #28301862, #28301870, … |
| maximal quotient of | #388×#6157, #390×#6157, #391×#6157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20977);