Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | D4 |
| commutator | C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C4 | C23×C5 |
| socle | C22×C5 | C22⋊C4 |
| prime radical | C2×C5×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ G | |
170 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(3,4,5,6,7);(8,9,10,13,11,14,15,12);(8,9)(10,13)(11,14)(12,15)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 921);