Finite Group DatabasebetaFinite Group Databasebeta| order | 1215 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C5 · Schur multiplier (H2) = C35 |
| direct product | C5×3−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C5 | C34 |
| commutator, Frattini, cosocle | C3 | C34×C5 |
| prime radical | C3×C5×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C5 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 135 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4,5)(6,7,10,9,13,20,18,12,24)(8,14,25,17,23,19,16,11,21)(15,27,31,28,22,32,29,26,30);(6,8,15)(7,11,22)(9,17,28)(10,19,30)(12,23,27)(13,14,26)(16,29,18)(20,21,31)(24,25,32);(8,16,17)(11,23,14)(15,28,29)(19,25,21)(22,26,27)(30,31,32);(7,12,13)(10,20,24)(11,23,14)(19,21,25)(22,27,26)(30,31,32)415 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1215, 66);