Finite Group DatabasebetaFinite Group Databasebeta| order | 1215 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C5 · Schur multiplier (H2) = C33 |
| direct product | C5×#6891 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C5 | C33 |
| commutator, Frattini, cosocle | C32 | C33×C5 |
| prime radical | C34×C5 | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C5 ◁ G | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5)(6,7,9,8,11,13,12,10,14)(15,16,18)(17,20,22)(19,23,21);(7,10,11)(9,13,14);(16,19,20)(18,22,23)255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1215, 38);