Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q32×C89 ⋊ C2 |
| Sylow | C89 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8×D89 |
| commutator | C8×C89 | C23 |
| Frattini | C8 | C22×D89 |
| Fitting | Q32×C89 | C2 |
| socle | C2×C89 | C2×D8 |
| hypercenter | Q32 | D89 |
| hypocenter | C89 | #508 |
| LCS | G ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32 | |
84 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|c^2=f,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^88,d^b=de,e^b=ef,g^b=g^88,g^c=g^88506 conjugacy classes — view the full character table.
| maximal subgroup of | #26022560, #26022562, #26022566, … |
| maximal normal subgroup of | #26022560, #26022562, #26022566, … |
| maximal quotient of | #26021958, #26022033, #26022098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 982);