Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C89 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #15667953 |
| commutator | C16×C89 | C22 |
| Frattini | C16 | C2×D89 |
| Fitting | Q32×C89 | C2 |
| socle | C2×C89 | D16 |
| hypercenter | Q32 | D89 |
| hypocenter | C89 | Q64 |
| LCS | G ▷ C16×C89 ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
30 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,c^b=cd,d^b=de,e^b=ef,g^b=g^88503 conjugacy classes — view the full character table.
| maximal subgroup of | Q64×D89, C2×#19151438, #26022559, … |
| maximal normal subgroup of | Q64×D89, C2×#19151438, #26022559, … |
| maximal quotient of | #26018432, #26018433, #26018451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 389);