Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #6728⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×S5 |
| commutator, hypocenter | SL(2,𝔽5) | C2×C8 |
| Fitting, hypercenter | C2×C8 | S5 |
| socle | C22 | C4×S5 |
| prime radical | C2×SL(2,𝔽5) | C2×C4 |
| cosocle | C4×SL(2,𝔽5) | C22 |
| LCS | G ▷ SL(2,𝔽5) | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
174 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Permutation generators:
(1,6,4,7,2,5,3,8)(9,31,20,22,26,14,12,29,18,23,27,15,10,32,19,21,25,13,11,30,17,24,28,16)(33,38,36,39,34,37,35,40)(41,64,52,53,58,45,44,62,50,56,59,48,42,63,51,54,57,46,43,61,49,55,60,47);(1,38)(2,37)(3,40)(4,39)(5,35)(6,36)(7,34)(8,33)(9,55,27,50,29,48,10,56,28,49,30,47)(11,54,26,52,31,45,12,53,25,51,32,46)(13,42,21,60,19,62,14,41,22,59,20,61)(15,44,23,57,18,64,16,43,24,58,17,63)72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240681);