Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C4×C5 · Schur multiplier (H2) = C32 |
| direct product | C4×C5×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5×C9 | C32 |
| commutator | C3 | C33×C4×C5 |
| Frattini, cosocle | C2×C3 | C2×C33×C5 |
| socle | C2×C3×C5 | C2×C33 |
| prime radical | C2×C5×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C4×C5×C9 ◁ G | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 540 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,7,8,9)(10,11,14)(12,16,22)(13,17,21)(15,23,20)(18,25,30)(19,26,29)(24,31,28)(27,33,35)(32,36,34);(11,15,17)(14,21,23)(16,24,26)(22,29,31)(25,32,33)(30,35,36);(10,12,18,13,19,27,20,28,34)(11,16,25,17,26,33,15,24,32)(14,22,30,21,29,35,23,31,36)660 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 261);