Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×D8×C89 ⋊ C2 |
| Sylow | C89 ⋊ #496 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D89 |
| commutator | C2×C4×C89 | C23 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | C2×D8×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | C2×D8 | D89 |
| hypocenter | C89 | #496 |
| LCS | G ▷ C2×C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
144 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,89|b^2=d,c^2=e,d^2=f,b^a=bd,c^a=ce,d^a=df,c^b=cf,g^b=g^88,g^c=g^88632 conjugacy classes — view the full character table.
| maximal subgroup of | #496×D89, C2×#19151986, #26030694, … |
| maximal normal subgroup of | #496×D89, C2×#19151986, #26030694, … |
| maximal quotient of | #26019873, #26019875, #26019878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 937);