Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C89 ⋊ #397 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D89 |
| commutator, socle | C23×C89 | C23 |
| Fitting | C2×C22⋊C4×C89 | C2 |
| hypercenter | C2×C22⋊C4 | D89 |
| hypocenter | C89 | #397 |
| LCS | G ▷ C23×C89 ▷ C89 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
170 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,89|a^2=e,c^2=d,b^a=bd,c^a=ce,c^b=cf,g^c=g^88902 conjugacy classes — view the full character table.
| maximal subgroup of | #26018251, #26018253, #26019238, … |
| maximal normal subgroup of | #26018251, #26018253, #26019238, … |
| maximal quotient of | #26018497, #26018499, #26018508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 474);