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) = C32 |
| direct product | C5×H(3,𝔽3)⋊C9 |
| unambiguous semidirect product | H(3,𝔽3)⋊(C5×C9) |
| unambiguous semidirect product | (C5×H(3,𝔽3))⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C5 | H(3,𝔽3) |
| commutator | C32 | C3×C5×C9 |
| Frattini, cosocle | C33 | C32×C5 |
| prime radical | C3×C5×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C5 ◁ C33×C5 ◁ G | |
94 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)(15,16,19,17,21,20,18,22,23);(15,16,20)(17,21,23)(18,22,19)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1215, 17);