Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×M16×C89 ⋊ C2 |
| Sylow | C89 ⋊ #407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D89 |
| commutator, socle | C22×C89 | C22×C4 |
| Fitting | C2×M16×C89 | C2 |
| hypercenter | C2×M16 | D89 |
| hypocenter | C89 | #407 |
| LCS | G ▷ C22×C89 ▷ C89 | |
| UCS | C1 ◁ C2×C4 ◁ C2×M16 | |
190 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|a^2=e,e^2=f,b^a=bd,c^a=cf,g^b=g^88908 conjugacy classes — view the full character table.
| maximal subgroup of | #26018187, #26020171, #26020174, … |
| maximal normal subgroup of | #26018187, #26020171, #26020174, … |
| maximal quotient of | #26018578, #26018581, #26018584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 512);