Finite Group DatabasebetaFinite Group Databasebeta| order | 5696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C89 ⋊ #475 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C22⋊D89 |
| commutator | C2×C4×C89 | C23 |
| Frattini | C2×C4 | C22×D89 |
| Fitting | C4×Q8×C89 | C2 |
| socle | C22×C89 | C2×D4 |
| hypercenter | C4×Q8 | D89 |
| hypocenter | C89 | #475 |
| LCS | G ▷ C2×C4×C89 ▷ C2×C89 ▷ C89 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×Q8 | |
96 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,89|a^2=d,b^2=d,c^2=f,e^2=f,b^a=bd,c^a=ce,e^a=ef,g^a=g^88,e^c=ef899 conjugacy classes — view the full character table.
| maximal subgroup of | #26019438, #26019441, #26019452, … |
| maximal normal subgroup of | #26019438, #26019441, #26019452, … |
| maximal quotient of | #26020257, #26020520, #26020830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5696, 834);