Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#15668036 |
| Fitting | C2×D8×C89 ⋊ C2 |
| Sylow | C89 ⋊ C2×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D89 |
| commutator | C4×C89 | C24 |
| Frattini | C4 | C23×D89 |
| Fitting | C2×D8×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | C2×D8 | D89 |
| hypocenter | C89 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
638 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#15668036, #26029063, #26029077, … |
| maximal normal subgroup of | C4×#15668036, #26029063, #26029077, … |
| maximal quotient of | C2×#19151601, C2×#19151596, C2×#19151639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 1334);