Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = #72277755 (order 26244) · Inn(G) = C32 · Out(G) = #2722 |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C3×C9 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C32 |
| commutator | C3 | C32×C9 |
| Frattini, socle, cosocle | C32 | C33 |
| prime radical | C33 | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3,5,7,6,8,9)(10,11,14)(12,16,22)(13,17,21)(15,23,20)(18,25,30)(19,26,29)(24,31,28)(27,33,35)(32,36,34);(11,15,17)(14,21,23)(16,24,26)(22,29,31)(25,32,33)(30,35,36);(10,12,18,13,19,27,20,28,34)(11,16,25,17,26,33,15,24,32)(14,22,30,21,29,35,23,31,36)Polycyclic presentation:
3^5|a^3=e,c^3=d,b^a=bd99 conjugacy classes — view the full character table.
| derived subgroup of | #92294⋊C2, #92295⋊C4, #92294⋊C4, … |
| Frattini subgroup of | #14546377, #14546378, #14546379, … |
| Fitting subgroup of | #92294, #92295, #92296, … |
| maximal subgroup of | #92294, #92295, #92296, … |
| maximal normal subgroup of | #92294, #92295, #92296, … |
| maximal quotient of | C2×#6887, #10624544, C9×C32⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 34);