Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | Aut(G) = C3×C3⋊(C3⋊#1088) (order 2916) · Inn(G) = H(3,𝔽3) · Out(G) = C3×C3⋊Dic3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | H(3,𝔽3) |
| commutator, Frattini, prime radical, cosocle | C33 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
35 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,4,6)(10,11,14,13,16,15,18,12,17);(1,2,5)(3,6,9)(4,8,7)(10,12,14,18,16,17,13,11,15)Polycyclic presentation:
3^5|a^3=d,b^3=d^2,b^a=bc,c^a=cd,c^b=ce35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 7);