Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | 3−1+4 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C3 | C32×C3⋊D3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C32 |
| Fitting | 3−1+4 | C2 |
| prime radical | C3×#272 | C3 |
| hypercenter | 3−1+2 | C3⋊D3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ 3−1+2 | |
184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,5,15,4,12,6,11,2,13)(3,22,23,19,18,24,9,17,21,8,14,27,10,25,16,20,7,26);(2,5,12)(3,9,10)(6,15,13)(8,19,20)(16,21,23)(17,25,22);(1,3,8)(2,7,17)(4,10,19)(5,14,22)(6,16,24)(9,20,11)(12,18,25)(13,21,26)(15,23,27)Polycyclic presentation:
2,3^5|c^3=f,d^a=d^2,e^a=e^2,c^b=cf,e^d=ef58 conjugacy classes — view the full character table.
| maximal subgroup of | #66419⋊C3, (#4207⋊C2)⋊C3, C2×#92394, … |
| maximal normal subgroup of | #66419⋊C3, (#4207⋊C2)⋊C3, C2×#92394, … |
| maximal quotient of | #11758015, C2×#92394, #13091378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 239);