Finite Group DatabasebetaFinite Group Databasebeta| order | 1053 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C13 · Schur multiplier (H2) = C32 |
| direct product | C13×C32⋊C9 |
| unambiguous semidirect product | C32⋊(C9×C13) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C13 | C32 |
| commutator | C3 | C3×C9×C13 |
| Frattini, cosocle | C32 | C32×C13 |
| prime radical | C33×C13 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C13 ◁ G | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 351 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17)(16,19,21)(18,22,20)(23,24,26,25,27,29,28,30,31);(15,18,19)(17,21,22)429 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1053, 3);