Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C33 |
| direct product | C2×C3×#6862 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C2×C33 |
| prime radical | C2×C34 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C2×C34 ◁ G | |
284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 486 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2)(3,4,5);(6,7,10)(8,14,18)(9,13,19)(11,20,16)(12,23,17)(15,25,28)(21,31,32)(22,30,27)(24,29,26)(33,34,37,44,50,56,36,39,46)(35,40,51,42,49,45,43,38,47)(41,48,55,54,52,57,53,58,59);(6,8,15,9,16,26,17,27,32)(7,11,21,13,22,25,12,14,24)(10,18,28,19,20,29,23,30,31)(33,35,41)(34,38,48)(36,43,53)(37,45,55)(39,49,58)(40,52,50)(42,54,44)(46,51,59)(47,57,56)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 146);