Finite Group DatabasebetaFinite Group Databasebeta| order | 8451 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C32⋊(C313⋊C3) |
| unambiguous semidirect product | C313⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×C313⋊C3)⋊C3 |
| Fitting | C32×C313 ⋊ C3 |
| Sylow | C313 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C313⋊C3 |
| commutator, socle | C3×C313 | C32 |
| Fitting | C32×C313 | C3 |
| hypercenter | C32 | C313⋊C3 |
| hypocenter | C313 | H(3,𝔽3) |
| LCS | G ▷ C3×C313 ▷ C313 | |
| UCS | C1 ◁ C3 ◁ C32 | |
22 conjugacy classes of subgroups.
Polycyclic presentation:
3^3,313|b^a=bc,d^b=d^214947 conjugacy classes — view the full character table.
| maximal subgroup of | #38761684, #38761685, #38761687, … |
| maximal normal subgroup of | #38761684, #38761685, #38761687, … |
| maximal quotient of | (C3×C313⋊C3)⋊C9, C32⋊(C313⋊C9), #38761683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8451, 9);