Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C36 |
| direct product | C2×C3×#6908 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C2×C34 |
| socle | C2×C32 | C3×H(3,𝔽3) |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C2×C32×C9 ◁ G | |
1064 conjugacy classes of subgroups; only the 476 normal subgroups are shown.
This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4,5);(6,7,10)(8,12,17)(9,11,18)(13,22,16)(14,20,25)(15,19,26)(21,29,24)(23,27,31)(28,32,30);(7,11,13)(12,19,21)(20,27,28);(6,8,14,9,15,23,16,24,30)(7,12,20,11,19,27,13,21,28)(10,17,25,18,26,31,22,29,32)306 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | C2×C3⋊#92330, C2×C3⋊#92280, C2×C3×#92344 |
| maximal subgroup of | C2×#14554857, C2×C3×#10624934, C2×C3⋊#92330, … |
| maximal normal subgroup of | C2×#14554857, C2×C3×#10624934, C2×C3⋊#92330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 483);