Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #6878 |
| automorphism | Aut(G) = C3×#4179⋊C3 (order 1458) · Inn(G) = #732 · Out(G) = C2×C32 |
| 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 | #732 | C3 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C32 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
22 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)(3,10,14)(4,9,15)(6,16,11)(7,20,12)(8,21,13)(17,25,23)(18,26,27)(19,24,22);(2,6,17,9,19,10,8,18,7)(5,14,16,21,25,26,15,20,24)Polycyclic presentation:
3^5|b^3=d^2,c^3=e^2,b^a=bc,c^a=cd,d^a=de35 conjugacy classes — view the full character table.
| Fitting subgroup of | #6878⋊C2 |
| maximal subgroup of | #6878⋊C2, C2×#6878, C3×#6878, … |
| maximal normal subgroup of | #6878⋊C2, C2×#6878, C3×#6878, … |
| maximal quotient of | C2×#6878, #10624481, #10624483, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 25);