Finite Group DatabasebetaFinite Group Databasebeta| order | 7533 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | #11719687⋊C9 |
| Fitting | C31×C9⋊C9 ⋊ C3 |
| Sylow | C31 ⋊ C9×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C32×C31⋊C3 |
| commutator | C3×C31 | C32×C9 |
| Fitting | C31×C9⋊C9 | C3 |
| socle | C32×C31 | C33 |
| hypercenter | C9⋊C9 | C31⋊C3 |
| hypocenter | C31 | C9×3−1+2 |
| LCS | G ▷ C3×C31 ▷ C31 | |
| UCS | C1 ◁ C32 ◁ C9⋊C9 | |
148 conjugacy classes of subgroups.
Polycyclic presentation:
3^5,31|a^3=e,b^3=d,c^3=e,b^a=bd,f^c=f^25429 conjugacy classes — view the full character table.
| maximal subgroup of | C9×3−1+2×C31⋊C3, C3×#11719687⋊C9, #36382261, … |
| maximal normal subgroup of | C9×3−1+2×C31⋊C3, C3×#11719687⋊C9, #36382261, … |
| maximal quotient of | #36379838, #36379842, C9⋊C9×C31⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7533, 170);