Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5×C9 · Schur multiplier (H2) = C3 |
| direct product | C4×C5×C9⋊C9 |
| unambiguous semidirect product | (C4×C9)⋊(C5×C9) |
| unambiguous semidirect product | (C4×C5×C9)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4×C5 | C32 |
| commutator | C3 | C3×C4×C5×C9 |
| Frattini, cosocle | C2×C32 | C2×C32×C5 |
| socle, prime radical | C2×C32×C5 | C2×C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C4×C5 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1620 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4)(5,6,7,8,9)(11,13,16)(14,18,17)(19,20,22,21,23,25,24,26,27);(10,11,14,12,13,17,15,16,18)660 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 64);