Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C9 · Schur multiplier (H2) = C33 |
| direct product | C2×C9×#733 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | H(3,𝔽3) |
| commutator | C32 | C2×C32×C9 |
| Frattini, cosocle | C33 | C2×C33 |
| socle | C2×C32 | C3×H(3,𝔽3) |
| prime radical | C2×C33 | C33 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C2×C32×C9 ◁ G | |
274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 486 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2)(3,4,5,6,7,8,9,10,11);(12,13,16,15,19,25,23,18,29)(14,20,24,22,17,26,33,28,36)(21,31,34,32,30,35,38,27,37);(12,14,21,23,33,38,15,22,32)(13,17,27,18,20,30,19,28,31)(16,24,34,29,36,37,25,26,35)306 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 279);