Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C9 · Schur multiplier (H2) = C35 |
| direct product | C2×C3×#6903 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C27 | C32 |
| commutator | C3 | C2×C33×C9 |
| Frattini, cosocle | C9 | C2×C34 |
| socle | C2×C32 | C32×C9 |
| prime radical | C2×C3×H(3,𝔽3) | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C27 ◁ G | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 86 |
| transitive degree | μt(G) = 486 |
A minimal faithful permutation representation — 4 generators on 86 points:
(1,2)(3,4,5);(6,7,11,9,13,23,19,29,44,10,14,24,20,30,45,38,53,67,21,31,46,39,54,68,60,74,82)(8,12,22,17,27,42,36,51,65,18,28,43,37,52,66,59,73,81,16,26,41,35,50,64,58,72,80)(15,25,40,33,48,62,56,70,78,32,47,61,55,69,77,75,83,85,34,49,63,57,71,79,76,84,86);(6,8,15)(7,12,25)(9,17,33)(10,18,32)(11,22,40)(13,27,48)(14,28,47)(16,34,21)(19,36,56)(20,37,55)(23,42,62)(24,43,61)(26,49,31)(29,51,70)(30,52,69)(35,57,39)(38,59,75)(41,63,46)(44,65,78)(45,66,77)(50,71,54)(53,73,83)(58,76,60)(64,79,68)(67,81,85)(72,84,74)(80,86,82);(8,16,18)(12,26,28)(15,32,34)(17,35,37)(22,41,43)(25,47,49)(27,50,52)(33,55,57)(36,58,59)(40,61,63)(42,64,66)(48,69,71)(51,72,73)(56,75,76)(62,77,79)(65,80,81)(70,83,84)(78,85,86)594 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 477);