Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | Aut(G) = C32×#66432 (order 2916) · Inn(G) = H(3,𝔽3) · Out(G) = C2×C32×D3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C32 |
| prime radical | C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,4,3,5,8,7,6,9)(11,13,15);(2,5,6)(10,11,14)(12,15,18)(13,17,16)Polycyclic presentation:
3^5|a^3=d,b^a=bc,c^a=cd,c^b=ce35 conjugacy classes — view the full character table.
| derived subgroup of | #92199, #11757322, #92161⋊C2, … |
| Fitting subgroup of | #92161, #92162, #92199, … |
| maximal subgroup of | #92161, #92162, #92199, … |
| maximal normal subgroup of | #92161, #92162, #92199, … |
| maximal quotient of | C2×#6857, #10624449, #10624450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 4);