Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | (C3⋊D3)⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×(C3⋊D3)⋊C3)⋊C3 |
| Fitting, Sylow | C32⋊H(3,𝔽3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×(C3⋊D3)⋊C3 |
| commutator, cosocle | C33 | C2×C32 |
| Frattini, socle | C32 | C32×D3 |
| Fitting | C32⋊H(3,𝔽3) | C2 |
| hypercenter | C32 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C2×H(3,𝔽3) |
| LCS | G ▷ C33 ▷ C32 | |
| UCS | C1 ◁ C3 ◁ C32 | |
153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,7,5,2,3,12)(4,15,11,8,9,17)(6,16,13,14,10,18);(1,5,10)(2,7,17)(3,4,11)(6,13,9)(8,15,18)(12,14,16)Polycyclic presentation:
2,3^5|e^a=e^2,f^a=f^2,c^b=cd,e^b=ef42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 102);