Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#91143 |
| derived | C3×SL(2,𝔽5) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | ΓL(2,𝔽4) |
| commutator, hypocenter | C3×SL(2,𝔽5) | C4 |
| Fitting, socle | C22×C3 | S5 |
| prime radical, cosocle | C2×C3×SL(2,𝔽5) | C2 |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C22 | |
108 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Permutation generators:
(1,30,23,21,15,8,4,2)(3,14,7,24,25,22,12,6)(5,20,27,36,31,29,19,10)(9,40,39,38,32,28,18,17)(11,35,37,33,34,26,16,13)(41,43,44,42)(46,47);(1,18,9,4,7,3)(2,34,31,21,11,5)(6,22,35,28,17,13)(8,27,16)(10,20)(12,25,15,39,32,23)(14,26,40,38,33,24)(19,37,30)(29,36)(41,42,44,43)(45,46)42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 1687);