Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5×C32 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×C4⋊C32 |
| unambiguous semidirect product | C4⋊(C3×C5×C32) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C5×C16 | C22 |
| commutator | C2 | C2×C3×C5×C32 |
| Frattini, cosocle | C2×C16 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5×C16 ◁ G | |
124 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 1920 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3)(4,5,6,7,8)(10,12)(13,14,15,19,16,20,23,29,17,21,24,30,26,32,35,39,18,22,25,31,27,33,36,40,28,34,37,41,38,42,43,44);(9,10,11,12)1200 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 153);