Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | H(3,ℤ/4)×D53 |
| Fitting | C53×H(3,ℤ/4) ⋊ C2 |
| Sylow | C53 ⋊ C2×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C42×D53 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×H(3,ℤ/4) | C2 |
| socle | C2×C53 | C2×#95 |
| hypercenter | H(3,ℤ/4) | D53 |
| hypocenter | C53 | C2×H(3,ℤ/4) |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ H(3,ℤ/4) | |
404 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
616 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#1339, D53×#1340, #27294239, … |
| maximal normal subgroup of | D53×#1339, D53×#1340, #27294239, … |
| maximal quotient of | D53×#1274, #27273489, #27273492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4209);