Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C89 ⋊ #428 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D356 |
| commutator | C4×C89 | C22×C4 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | C2×M16×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | C2×M16 | D89 |
| hypocenter | C89 | #428 |
| LCS | G ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×M16 | |
118 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|a^2=e,b^2=d,d^2=f,b^a=bd,d^a=df,g^a=g^88,c^b=cf902 conjugacy classes — view the full character table.
| maximal subgroup of | #26020463, #26020464, #26020468, … |
| maximal normal subgroup of | #26020463, #26020464, #26020468, … |
| maximal quotient of | #26019004, #26019019, #26019037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 591);