Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | #97×C89 ⋊ C2 |
| Sylow | C89 ⋊ #497 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D356 |
| commutator | C2×C4×C89 | C23 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | #97×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | #97 | D89 |
| hypocenter | C89 | #497 |
| LCS | G ▷ C2×C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #97 | |
124 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|a^2=f,b^2=d,c^2=e,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=g^88,c^b=cf896 conjugacy classes — view the full character table.
| maximal subgroup of | #26019560, #26019565, #26019571, … |
| maximal normal subgroup of | #26019560, #26019565, #26019571, … |
| maximal quotient of | #26019874, #26019885, #26019898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 943);