Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | Aut(G) = C3×#14519523 (order 5832) · Inn(G) = C33 · Out(G) = C3×#626 |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C9 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C32 | C33 |
| prime radical | C33 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,6,8)(3,7,9)(11,12,16)(14,17,15);(1,2,7,5,8,9,4,6,3)(10,11,14,13,12,15,18,16,17);(1,3,6,4,9,8,5,7,2)(10,12,17,13,16,14,18,11,15)Polycyclic presentation:
3^5|b^3=e,c^3=d,b^a=bd,c^a=ce51 conjugacy classes — view the full character table.
| derived subgroup of | #11757690, #92308⋊C2, #14516733, … |
| Fitting subgroup of | #92264, #92308, #92309, … |
| maximal subgroup of | #92264, #92308, #92309, … |
| maximal normal subgroup of | #92264, #92308, #92309, … |
| maximal quotient of | C2×#6896, #10624550, #10624557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 43);