Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of C5⋊#4175 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C5⋊#4175 |
| Fitting, Hall | C5×AΣL(1,𝔽27) ⋊ C2 |
| derived | C33×C5 ⋊ C2×C3 |
| Sylow | AΣL(1,𝔽27) ⋊ D5 |
| Sylow | C5 ⋊ #4175 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×C5 | C2×C3 |
| Frattini | C32 | C3×D15 |
| Fitting | C5×AΣL(1,𝔽27) | C2 |
| socle | C3×C5 | (C3⋊D3)⋊C3 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 45 |
Polycyclic presentation:
2,3^4,5|c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^4,c^b=cd,d^b=de47 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#4175, #4174⋊D5, #4183⋊D5, … |
| maximal normal subgroup of | D5×#4175, #4174⋊D5, #4183⋊D5, … |
| maximal quotient of | #13108137, C2×C5⋊#4175 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 35);