Finite Group DatabasebetaFinite Group Databasebeta| order | 1350 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C52 · Schur multiplier (H2) = C5 |
| direct product | C2×C52×3−1+2 |
| unambiguous semidirect product | (C2×C9)⋊(C3×C52) |
| unambiguous semidirect product | (C2×C5×C9)⋊(C3×C5) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C52 | C32 |
| commutator, Frattini, cosocle | C3 | C2×C32×C52 |
| prime radical | C2×C32×C52 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C52 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 450 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(3,4,5,6,7)(13,14,16,15,18,20,19,17,21);(8,9,10,11,12)(14,17,18)(16,20,21)550 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1350, 91);