Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C3×(#4178⋊C2)⋊C3 (order 2916) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C32×D3 |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C33 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C9 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C33 |
| socle | C3 | C3×H(3,𝔽3) |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,5)(3,7,12)(4,6,13)(8,17,11)(9,15,20)(10,14,21)(16,24,19)(18,22,26)(23,27,25);(2,6,8)(7,14,16)(15,22,23);(1,3,9,4,10,18,11,19,25)(2,7,15,6,14,22,8,16,23)(5,12,20,13,21,26,17,24,27)Polycyclic presentation:
3^5|c^3=e,b^a=bd,d^a=de51 conjugacy classes — view the full character table.
| derived subgroup of | #92344, #11757648, #92280⋊C2, … |
| Fitting subgroup of | #92280, #92330, #92344, … |
| maximal subgroup of | #92280, #92330, #92344, … |
| maximal normal subgroup of | #92280, #92330, #92344, … |
| maximal quotient of | C2×#6908, #10624583, #10624585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 55);