Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C36 |
| direct product | C2×H(3,𝔽3)×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C34 |
| commutator, Frattini, cosocle | C32 | C2×C34 |
| prime radical | C2×C32×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
1244 conjugacy classes of subgroups; only the 518 normal subgroups are shown.
This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 518 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4,6)(5,8,10)(7,11,9);(4,7,8)(6,10,11);(12,13,15,14,17,19,18,16,20);(13,16,17)(15,19,20)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 455);