Finite Group DatabasebetaFinite Group Databasebeta| order | 1215 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6875 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×#6875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C5 | C9⋊C9 |
| commutator | C9 | C3×C5×C9 |
| Frattini, cosocle | C3×C9 | C32×C5 |
| prime radical | C32×C5 | 3−1+2 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C5 ◁ C3×C5×C9 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,4,5)(7,10,16,12,19,28,23,31,18)(8,13,24)(11,20,17,21,32,29,22,27,30)(14,26,25);(6,7,11,8,10,17,14,18,20,9,12,22,13,19,30,25,16,27,15,23,21,24,31,29,26,28,32)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1215, 22);