Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of AGL(1,𝔽25) |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #6053⋊C9 |
| derived, Sylow | C52 ⋊ C8×C9 |
| Hall | C52⋊C9 ⋊ C8 |
| Hall | #6053 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | AGL(1,𝔽25) |
| commutator, hypocenter | C52 | C8×C9 |
| Fitting, socle | C3×C52 | C3×C8 |
| prime radical | C3×C5⋊D5 | C3×C4 |
| cosocle | C3×#1043 | C2×C3 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C3 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 90 |
Polycyclic presentation:
2,3,2,3,2,5^2|a^2=c,b^3=d,c^2=e,f^a=f^3g^3,g^a=f^3g^2,f^b=g^3,g^b=f^3g^4,f^c=f^3,g^c=g^3,f^e=f^4,g^e=g^475 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 271);