Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×C5×#103 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32×C5 | D4 |
| commutator | C4 | C2×C32×C4×C5 |
| Frattini, cosocle | C2×C4 | C22×C32×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C5 ◁ C2×C32×C4×C5 ◁ G | |
252 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 1440 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(7,8,9,10,11)(12,13,14,15)(17,20)(18,22)(21,23);(4,5,6)(16,17,19,21)(18,23,22,20)630 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 1446);