Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #92319 (order 486) · Inn(G) = C32 · Out(G) = D3×C9 |
| homology | Abelianization (H1) = C3×C27 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C81⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C27 | C32 |
| commutator, socle | C3 | C3×C27 |
| prime radical | C32 | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C27 ▷ C9 ▷ C3 ▷ C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,2,6,3,8,18,11,23,37,4,9,19,12,24,38,28,45,59,14,26,39,30,47,60,50,66,74,5,10,17,13,25,35,29,46,56,15,27,36,31,48,57,51,67,72,33,49,58,53,68,73,69,78,80,16,7,20,32,21,40,52,42,61,34,22,41,54,43,62,70,64,75,55,44,63,71,65,76,79,77,81);(2,7,10)(6,17,20)(8,21,25)(9,22,27)(18,35,40)(19,36,41)(23,42,46)(24,43,48)(26,44,49)(37,56,61)(38,57,62)(39,58,63)(45,64,67)(47,65,68)(59,72,75)(60,73,76)(66,77,78)(74,80,81)Polycyclic presentation:
3^5|a^3=c,c^3=d,d^3=e,b^a=be99 conjugacy classes — view the full character table.
| Fitting subgroup of | D81⋊C3 |
| maximal subgroup of | D81⋊C3, C2×C81⋊C3, #10624500, … |
| maximal normal subgroup of | D81⋊C3, C2×C81⋊C3, #10624500, … |
| maximal quotient of | C2×C81⋊C3, #10624499, C32⋊C81, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 24);