Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | Aut(G) = C3×#13091033 (order 4374) · Inn(G) = C33 · Out(G) = C2×AΣL(1,𝔽27) |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C3 |
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 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,5,3)(6,7,8)(10,11,15)(12,19,31)(13,17,25)(14,18,26)(16,27,22)(20,30,34)(21,32,24)(23,29,35)(28,36,33);(1,2,6,9,3,7,4,5,8)(10,12,20,14,21,33,23,22,13)(11,16,28,18,19,17,29,32,30)(15,24,25,26,27,34,35,31,36);(1,3,8,9,5,6,4,2,7)(10,13,22,23,33,21,14,20,12)(11,17,16,29,28,32,18,30,19)(15,25,27,35,36,24,26,34,31)Polycyclic presentation:
3^5|b^3=de,c^3=d^2,b^a=bd,c^a=ce51 conjugacy classes — view the full character table.
| Fitting subgroup of | #6899⋊C2 |
| maximal subgroup of | #6899⋊C2, C2×#6899, #10624853, … |
| maximal normal subgroup of | #6899⋊C2, C2×#6899, #10624853, … |
| maximal quotient of | C2×#6899, #10624553, #10624557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 46);