Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C89 ⋊ #499 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D89 |
| commutator | C2×C4×C89 | C23 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | #106×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | #106 | D89 |
| hypocenter | C89 | #499 |
| LCS | G ▷ C2×C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #106 | |
86 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|a^2=d,b^2=df,c^2=e,e^2=f,b^a=bd,c^a=ce,e^a=ef,g^b=g^88638 conjugacy classes — view the full character table.
| maximal subgroup of | #26022426, #26022427, #499×D89, … |
| maximal normal subgroup of | #26022426, #26022427, #499×D89, … |
| maximal quotient of | #26020413, #26020978, #26020981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 949);