Finite Group DatabasebetaFinite Group Databasebeta| order | 1053 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C34 |
| direct product | H(3,𝔽3)×C13⋊C3 |
| Fitting | C13×H(3,𝔽3) ⋊ C3 |
| Sylow | C13 ⋊ C3×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×C13⋊C3 |
| commutator, socle, cosocle | C3×C13 | C33 |
| Fitting | C13×H(3,𝔽3) | C3 |
| hypercenter | H(3,𝔽3) | C13⋊C3 |
| hypocenter | C13 | C3×H(3,𝔽3) |
| LCS | G ▷ C3×C13 ▷ C13 | |
| UCS | C1 ◁ C3 ◁ H(3,𝔽3) | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 117 |
77 conjugacy classes — view the full character table.
| maximal subgroup of | C13⋊C3×AΣL(1,𝔽27), C13⋊C3×#731, C13⋊C3×#732, … |
| maximal normal subgroup of | C13⋊C3×AΣL(1,𝔽27), C13⋊C3×#731, C13⋊C3×#732, … |
| maximal quotient of | C13⋊C3×C32⋊C9, #15691557, H(3,𝔽3)×C13⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1053, 40);