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) = C34 |
| direct product | C2×C3×#6894 |
| 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 | |
932 conjugacy classes of subgroups; only the 548 normal subgroups are shown.
This group has 932 conjugacy classes of subgroups — too many to draw. Only its 548 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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);(7,9,12)(10,14,13)(15,16,18,17,20,22,21,19,23);(6,7,10,8,9,13,11,12,14);(16,19,20)(18,22,23)306 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 430);