Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C33×C9 |
| direct product | C2×C3×#6896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | C33 |
| commutator, Frattini, cosocle | C32 | C2×C34 |
| prime radical | C2×C34 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ G | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 486 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(3,4,5);(7,11,13)(8,12,14)(16,17,21)(19,22,20);(6,7,12,10,13,14,9,11,8)(15,16,19,18,17,20,23,21,22);(6,8,11,9,14,13,10,12,7)(15,17,22,18,21,19,23,16,20)306 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 433);