Finite Group DatabasebetaFinite Group Databasebeta| order | 1350 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C25 · Schur multiplier (H2) = C1 |
| direct product | C2×C25×3−1+2 |
| unambiguous semidirect product | (C2×C9)⋊(C3×C25) |
| unambiguous semidirect product | (C2×C9×C25)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C25 | C32 |
| commutator | C3 | C2×C32×C25 |
| Frattini, cosocle | C3×C5 | C2×C32×C5 |
| socle | C2×C3×C5 | C32×C5 |
| prime radical | C2×C32×C5 | C3×C5 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C25 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 450 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,31,30,33,35,34,32,36);(29,32,33)(31,35,36)550 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1350, 22);