Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, Sylow | C32⋊H(3,𝔽3) ⋊ C2 |
| derived | C3×H(3,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C3×H(3,𝔽3) | C2×C3 |
| Frattini, socle | C32 | C3×C3⋊D3 |
| Fitting | C32⋊H(3,𝔽3) | C2 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,15,11,8,9,16)(4,12)(5,7)(6,18,14,10,13,17);(3,9,11)(6,14,13)(8,16,15)(10,18,17);(1,3,10)(2,6,8)(4,9,17)(5,11,18)(7,13,16)(12,14,15)Polycyclic presentation:
2,3^5|c^a=c^2,d^a=d^2,e^a=e^2,c^b=ce,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | (((C3⋊D3)⋊C3)⋊D3)⋊C3, #11757730, #11757749, … |
| maximal normal subgroup of | (((C3⋊D3)⋊C3)⋊D3)⋊C3, #11757730, #11757749, … |
| maximal quotient of | #11757604, C2×#92300, #13090620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 145);