Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×#13083563 |
| Fitting | C4×C8×C89 ⋊ C2 |
| Sylow | C89 ⋊ C4×SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D356 |
| commutator | C4×C89 | C22×C4 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | C4×C8×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | C4×C8 | D89 |
| hypocenter | C89 | C4×SD16 |
| LCS | G ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ C4×C8 | |
122 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1436 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #26019051, C8×#13083563, #26019063, … |
| maximal normal subgroup of | #26019051, C8×#13083563, #26019063, … |
| maximal quotient of | C8×#13083563, #26019090, #26019097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 639);