Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | M32×C89 ⋊ C2 |
| Sylow | C89 ⋊ #510 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D712 |
| commutator | C8×C89 | C23 |
| Frattini | C8 | C22×D89 |
| Fitting | M32×C89 | C2 |
| socle | C2×C89 | C2×D8 |
| hypercenter | M32 | D89 |
| hypocenter | C89 | #510 |
| LCS | G ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ M32 | |
82 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|b^2=f,d^2=ef,e^2=f,b^a=bd,c^a=cf,d^a=de,e^a=ef,g^a=g^88,d^b=de,e^b=ef,g^b=g^88896 conjugacy classes — view the full character table.
| maximal subgroup of | #26022142, #26022145, #26022204, … |
| maximal normal subgroup of | #26022142, #26022145, #26022204, … |
| maximal quotient of | #26021965, #26021973, #26022040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 992);