Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊#4174) |
| Fitting, Sylow | C32×AΣL(1,𝔽27) ⋊ C2 |
| derived | C32×H(3,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| commutator, cosocle, hypocenter | C32×H(3,𝔽3) | C2×C3 |
| Frattini | C32 | C3×C3⋊(C3⋊D3) |
| Fitting | C32×AΣL(1,𝔽27) | C2 |
| socle | C33 | (C3⋊D3)⋊C3 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
1100 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2,3^6|b^a=b^2,d^a=d^2,e^a=e^2,f^a=f^2g,c^b=cf,f^b=fg90 conjugacy classes — view the full character table.
| maximal subgroup of | #72277644 |
| maximal normal subgroup of | #72277644 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 1649);