Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×D8×C89 ⋊ C2 |
| Sylow | C89 ⋊ #472 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #15667983 |
| commutator | C2×C4×C89 | C23 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | C2×D8×C89 | C2 |
| socle | C2×C89 | #121 |
| hypercenter | C2×D8 | D89 |
| hypocenter | C89 | #472 |
| LCS | G ▷ C2×C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×D8 | |
108 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|b^2=df,d^2=f,e^2=f,b^a=bd,c^a=ce,d^a=df,e^a=ef,e^c=ef,g^c=g^88632 conjugacy classes — view the full character table.
| maximal subgroup of | #26022206, #26022208, #26022213, … |
| maximal normal subgroup of | #26022206, #26022208, #26022213, … |
| maximal quotient of | #26019604, #26019609, #26019648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 816);