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) = C34 |
| direct product | C2×C3×#6867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | H(3,𝔽3) |
| commutator | C32 | C2×C32×C9 |
| Frattini, cosocle | C33 | C2×C33 |
| prime radical | C2×C34 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C2×C34 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 486 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2)(3,4,5);(6,7,9,8,10,12,11,13,14)(15,16,19)(17,23,27)(18,22,28)(20,29,25)(21,32,26)(24,34,37)(30,40,41)(31,39,36)(33,38,35);(15,17,24,18,25,35,26,36,41)(16,20,30,22,31,34,21,23,33)(19,27,37,28,29,38,32,39,40)306 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 246);