Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | #107×C89 ⋊ C2 |
| Sylow | C89 ⋊ #482 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D89 |
| commutator | C2×C4×C89 | C23 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | #107×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | #107 | D89 |
| hypocenter | C89 | #482 |
| LCS | G ▷ C2×C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ #107 | |
106 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 37 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=f,c^2=d,e^2=f,b^a=bd,c^a=ce,e^a=ef,g^a=g^88,g^b=g^88,e^c=ef,g^c=g^88635 conjugacy classes — view the full character table.
| maximal subgroup of | #482×D89, C2×#19151925, #26030168, … |
| maximal normal subgroup of | #482×D89, C2×#19151925, #26030168, … |
| maximal quotient of | #26020346, #26020448, #26020573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 876);