Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | Aut(G) = C9×((C3⋊D3)⋊C3)⋊C2 (order 972) · Inn(G) = H(3,𝔽3) · Out(G) = C22×C9 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C9 | H(3,𝔽3) |
| commutator | C32 | C3×C9 |
| Frattini, cosocle | C3×C9 | C32 |
| socle | C3 | C32⋊C9 |
| prime radical | H(3,𝔽3) | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,2,6,4,9,21,16,27,43,5,10,20,17,28,42,38,52,65,18,8,24,39,26,48,62,51,69)(3,11,29,14,31,54,36,56,73,15,25,47,37,50,68,61,71,79,13,7,19,35,23,41,60,46,64)(12,30,40,33,55,63,58,74,77,32,49,53,57,70,72,75,80,81,34,22,44,59,45,66,76,67,78);(1,3,12)(2,7,22)(4,14,33)(5,15,32)(6,19,40)(8,25,49)(9,23,45)(10,11,30)(13,34,18)(16,36,58)(17,37,57)(20,29,53)(21,41,63)(24,47,44)(26,50,70)(27,46,67)(28,31,55)(35,59,39)(38,61,75)(42,54,72)(43,64,77)(48,68,66)(51,71,80)(52,56,74)(60,76,62)(65,73,81)(69,79,78)Polycyclic presentation:
3^5|a^3=d,d^3=e,b^a=bc,c^a=ce,c^b=ce51 conjugacy classes — view the full character table.
| derived subgroup of | #92213, #11757336, #92181⋊C2, … |
| Fitting subgroup of | #92181, #92182, #92213, … |
| maximal subgroup of | #92181, #92182, #92213, … |
| maximal normal subgroup of | #92181, #92182, #92213, … |
| maximal quotient of | C2×#6872, H(3,𝔽3)⋊C27, 3−1+2⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 19);