Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6875 |
| automorphism | Aut(G) = #6895⋊C2 (order 486) · Inn(G) = C9⋊C9 · Out(G) = D3 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C9⋊C9 |
| commutator | C9 | C3×C9 |
| Frattini, cosocle | C3×C9 | C32 |
| prime radical | C32 | 3−1+2 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,5,11,7,14,23,18,26,13)(3,8,19)(6,15,12,16,27,24,17,22,25)(9,21,20);(1,2,6,3,5,12,9,13,15,4,7,17,8,14,25,20,11,22,10,18,16,19,26,24,21,23,27)Polycyclic presentation:
3^5|a^3=d,b^3=c,c^3=e,b^a=bc,c^a=ce,d^b=de^235 conjugacy classes — view the full character table.
| Fitting subgroup of | AGL(1,ℤ/27) |
| maximal subgroup of | AGL(1,ℤ/27), C2×#6875, C3×#6875, … |
| maximal normal subgroup of | AGL(1,ℤ/27), C2×#6875, C3×#6875, … |
| maximal quotient of | C2×#6875, #10624462, #10624472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 22);