Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6909 |
| automorphism | Aut(G) = C3×#92333 (order 1458) · Inn(G) = C3×H(3,𝔽3) · Out(G) = C3×D3 |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C33 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C33 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,5)(3,7,12)(4,6,13)(8,17,11)(9,14,21)(10,16,22)(15,24,18)(19,23,26)(20,25,27);(2,6,8)(3,9,18)(7,15,14)(10,20,19)(12,21,24)(22,27,26);(1,3,10,4,9,19,11,18,20)(2,7,16,6,14,23,8,15,25)(5,12,22,13,21,26,17,24,27)Polycyclic presentation:
3^5|c^3=e,b^a=bd,d^a=de,c^b=ce35 conjugacy classes — view the full character table.
| Fitting subgroup of | #6913⋊C2 |
| maximal subgroup of | #6913⋊C2, C2×#6913, #10624851, … |
| maximal normal subgroup of | #6913⋊C2, C2×#6913, #10624851, … |
| maximal quotient of | C2×#6913, #10624595, #10624601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 60);