Finite Group DatabasebetaFinite Group Databasebeta| order | 1215 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C9 · Schur multiplier (H2) = C3 |
| direct product | C5×3−1+2⋊C9 |
| unambiguous semidirect product | 3−1+2⋊(C5×C9) |
| unambiguous semidirect product | (C5×3−1+2)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C5 | H(3,𝔽3) |
| commutator | C32 | C3×C5×C9 |
| Frattini, cosocle | C33 | C32×C5 |
| prime radical | C33×C5 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5 ◁ C33×C5 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 405 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5)(6,7,9,8,10,12,11,13,14)(15,16,19)(17,21,18)(20,23,22);(15,17,21,18,20,23,22,16,19)255 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×3−1+2⋊C9, C5×#10624697, C5×#10624699, … |
| maximal normal subgroup of | C3×C5×3−1+2⋊C9, C5×#10624697, C5×#10624699, … |
| maximal quotient of | C5×3−1+2⋊C27, C5×#10624450, C5×#10624451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1215, 18);