Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = D4×AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | SD16×AGL(1,𝔽5) |
| Fitting | C5×SD16 ⋊ C4 |
| Sylow | C5 ⋊ C4×SD16 |
122 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,3,4,5);(1,2,3,5,4)(7,10)(8,13)(11,12);(6,7,9,12)(8,10,13,11)35 conjugacy classes — view the full character table.
| maximal subgroup of | C2×SD16×AGL(1,𝔽5), #10617428, AGL(1,𝔽5)×#135, … |
| maximal normal subgroup of | C2×SD16×AGL(1,𝔽5), #10617428, AGL(1,𝔽5)×#135, … |
| maximal quotient of | #10603936, #10603941, #10603942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1072);