Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽3) |
| automorphism | Aut(G) order 8398080 |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C35 |
| extraspecial | 3+1+4 |
| Heisenberg | H(5,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C3 | C34 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,5)(3,9,20)(4,8,15)(6,16,11)(7,19,13)(10,22,26)(12,18,14)(17,27,23)(21,25,24);(1,3,10)(2,6,17)(4,12,23)(5,14,25)(7,18,22)(8,9,21)(11,24,13)(15,16,26)(19,20,27);(3,11,12)(6,18,9)(10,23,24)(14,20,16)(17,21,22)(25,26,27);(2,7,8)(5,15,19)(6,18,9)(14,16,20)(17,22,21)(25,26,27)Polycyclic presentation:
3^5|b^a=be,d^a=de,c^b=ce83 conjugacy classes — view the full character table.
| derived subgroup of | #92404, #11758025, #11758049, … |
| Fitting subgroup of | #92391, #92392, #92404, … |
| maximal subgroup of | #92391, #92392, #92404, … |
| maximal normal subgroup of | #92391, #92392, #92404, … |
| maximal quotient of | C2×H(5,𝔽3), #10624862, #10624880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 65);