Finite Group DatabasebetaFinite Group Databasebeta| order | 1053 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C13 · Schur multiplier (H2) = C3 |
| direct product | C13×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C13 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C32×C13 |
| prime radical | C13×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C13 ◁ C32×C13 ◁ G | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 351 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18)(16,22,26)(17,21,27)(19,28,24)(20,31,25)(23,33,36)(29,39,40)(30,38,35)(32,37,34);(14,16,23,17,24,34,25,35,40)(15,19,29,21,30,33,20,22,32)(18,26,36,27,28,37,31,38,39)221 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1053, 8);