Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×C5×#1319 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C5 | #99 |
| commutator | C2×C4 | C2×C3×C5×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C22×C3×C5 |
| prime radical | C23×C3×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C5 ◁ C23×C3×C5 ◁ C2×C3×C5×C4⋊C4 ◁ G | |
268 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 480 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3)(4,5,6,7,8)(9,10,11,13,12,14,15,16)(17,18)(19,25,27,32)(20,28,30,23)(21,24)(22,26)(29,31);(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25)390 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 51);