Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C3×#72277606 (order 34992) · Inn(G) = C32 · Out(G) = C3×#13049364 |
| homology | Abelianization (H1) = C92 · Schur multiplier (H2) = C33 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C33 | C32 |
| commutator | C3 | C92 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,13,12,14,16,15,17,18);(2,5,6)(4,8,9)(19,20,22,21,23,25,24,26,27)Polycyclic presentation:
3^5|a^3=d,b^3=e,b^a=bc99 conjugacy classes — view the full character table.
| derived subgroup of | #92197, #11757320, #11757369, … |
| Fitting subgroup of | #92158, #92197, #11757369, … |
| maximal subgroup of | #92158, #92197, C2×#6855, … |
| maximal normal subgroup of | #92158, #92197, C2×#6855, … |
| maximal quotient of | C2×#6855, #10624463, H(3,ℤ/9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 2);