Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #6878 |
| automorphism | Aut(G) = #72277466 (order 4374) · Inn(G) = #732 · Out(G) = C3×C3⋊D3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C32 |
| prime radical | C32 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
21 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,5,4,8,15,11,20,14)(3,7,12,10,17,13,22,27,24)(6,16,26,18,25,9,19,23,21);(2,6,17,8,18,27,20,19,7)(5,12,23,14,24,25,15,13,16)Polycyclic presentation:
3^5|a^3=e,b^3=d^2e,c^3=e^2,b^a=bc,c^a=cd,d^a=de35 conjugacy classes — view the full character table.
| Fitting subgroup of | #6880⋊C2 |
| maximal subgroup of | #6880⋊C2, C2×#6880, #10624536, … |
| maximal normal subgroup of | #6880⋊C2, C2×#6880, #10624536, … |
| maximal quotient of | C2×#6880, #10624482, #10624491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 27);