Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6909 |
| automorphism | Aut(G) = #72277588 (order 13122) · Inn(G) = C3×H(3,𝔽3) · Out(G) = #4175 |
| 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 |
| prime radical | C33 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,5,4,8,15,13,7,19)(3,9,14,12,6,16,11,18,25)(10,21,24,22,20,26,23,17,27);(1,3,10,13,11,23,4,12,22)(2,6,17,7,9,20,8,18,21)(5,14,24,19,25,27,15,16,26);(3,11,12)(6,9,18)(10,22,23)(14,25,16)(17,21,20)(24,26,27)Polycyclic presentation:
3^5|a^3=e,b^3=e^2,b^a=bd,d^a=de,c^b=ce35 conjugacy classes — view the full character table.
| Fitting subgroup of | #6912⋊C2 |
| maximal subgroup of | #6912⋊C2, C2×#6912, #10624750, … |
| maximal normal subgroup of | #6912⋊C2, C2×#6912, #10624750, … |
| maximal quotient of | C2×#6912, #10624592, #10624594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 59);