Finite Group DatabasebetaFinite Group Databasebeta| order | 1053 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C13⋊C9)⋊C9 |
| Sylow | C13 ⋊ C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C3×C13⋊C3 |
| commutator | C3×C13 | C3×C9 |
| Fitting | C3×C9×C13 | C3 |
| socle, prime radical, cosocle | C32×C13 | C32 |
| hypercenter | C3×C9 | C13⋊C3 |
| hypocenter | C13 | C9⋊C9 |
| LCS | G ▷ C3×C13 ▷ C13 | |
| UCS | C1 ◁ C32 ◁ C3×C9 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1053 |
Polycyclic presentation:
3^4,13|a^3=d,b^3=c,b^a=bc,e^b=e^3141 conjugacy classes — view the full character table.
| maximal subgroup of | C13⋊C3×C9⋊C9, C3×(C13⋊C9)⋊C9, #15691553, … |
| maximal normal subgroup of | C13⋊C3×C9⋊C9, C3×(C13⋊C9)⋊C9, #15691553, … |
| maximal quotient of | C13⋊#6855, (C13⋊C9)⋊C27, #15691509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1053, 20);