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