Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×ΓL(2,𝔽4) |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽5) | C22 |
| Fitting | C22×C3 | S5 |
| socle | C2×C3 | C2×S5 |
| hypercenter | C22 | ΓL(2,𝔽4) |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ C22 | |
150 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Permutation generators:
(1,2,8,25,20,30,13,3)(4,9,7,12,35,11,5,17)(6,21,29,14,27,16,36,19)(10,31,18,26,15,28,24,32)(22,40,48,39,44,42,43,34)(23,37,47,38,45,41,46,33)(50,51);(1,4,16,34,24,7,20,35,21,39,18,5)(2,9,25,19,38,15,30,11,3,14,33,10)(6,13,32,48,37,44,27,8,26,43,41,22)(12,31,47,42,45,29,17,28,46,40,23,36)(49,50)36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4638);