Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#15667996 |
| Fitting | C2×C22⋊C4×C89 ⋊ C2 |
| Sylow | C89 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D89 |
| commutator | C22×C89 | C24 |
| Frattini | C22 | C23×D89 |
| Fitting | C2×C22⋊C4×C89 | C2 |
| socle | C23×C89 | C23 |
| hypercenter | C2×C22⋊C4 | D89 |
| hypocenter | C89 | C2×#123 |
| LCS | G ▷ C22×C89 ▷ C89 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
908 conjugacy classes — view the full character table.
| maximal subgroup of | #26020664, #26020709, #26021382, … |
| maximal normal subgroup of | #26020664, #26020709, #26021382, … |
| maximal quotient of | C2×#19151459, C2×#19151473, C2×#19151475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 1053);