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×AΣL(1,𝔽27) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C13 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C32×C13 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C13 ◁ C32×C13 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 117 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17)(16,18,21)(19,22,20);(15,18,19)221 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1053, 7);