Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C3×#72243789 (order 8748) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C3×#272 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | H(3,𝔽3) |
| commutator | C32 | C3×C9 |
| Frattini, prime radical, cosocle | C33 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,4,3,5,7,6,8,9)(10,11,14)(12,18,22)(13,17,23)(15,24,20)(16,27,21)(19,29,32)(25,35,36)(26,34,31)(28,33,30);(10,12,19,13,20,30,21,31,36)(11,15,25,17,26,29,16,18,28)(14,22,32,23,24,33,27,34,35)Polycyclic presentation:
3^5|a^3=d,b^3=e,b^a=bc,c^a=ce51 conjugacy classes — view the full character table.
| derived subgroup of | #92208, #11757331, #92175⋊C2, … |
| Fitting subgroup of | #92175, #92176, #92208, … |
| maximal subgroup of | #92175, #92176, #92208, … |
| maximal normal subgroup of | #92175, #92176, #92208, … |
| maximal quotient of | C2×#6867, #10624445, #10624450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 14);