Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| isoclinism | isoclinic class of C3⋊(#4183⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊(#4183⋊C3) |
| Fitting, Sylow | C3×#6909 ⋊ C2 |
| derived | C3×AΣL(1,𝔽27) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AΣL(1,𝔽27) | C2×C3 |
| Frattini | C32 | C3×C3⋊(C3⋊D3) |
| Fitting | C3×#6909 | C2 |
| socle | C32 | C3×#272 |
| LCS | G ▷ C3×AΣL(1,𝔽27) | |
| UCS | C1 | |
563 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 563 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
2,3^6|b^a=b^2,c^a=c^2,e^a=e^2,f^a=fg^2,g^a=g^2,c^b=cf,f^b=fg,d^c=dg66 conjugacy classes — view the full character table.
| maximal subgroup of | #72221905, #72221961, #72221974, … |
| maximal normal subgroup of | #72221905, #72221961, #72221974, … |
| maximal quotient of | #72221909, #72259864, C2×C3⋊(#4183⋊C3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 1693);