Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×D5×D8 |
| Fitting | C2×C5×D8 ⋊ C2 |
| Sylow | C5 ⋊ C22×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C2×C5×D8 | C2 |
| socle | C22×C5 | C2×D4 |
| hypercenter | C2×D8 | D5 |
| hypocenter | C5 | C22×D8 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
338 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 4 generators on 15 points:
(1,2)(3,4,6,7,5)(8,9)(10,15)(11,13)(12,14);(1,2)(9,12)(10,14)(13,15);(4,5)(6,7);(1,2)56 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#357, #10599983, #10599984, … |
| maximal normal subgroup of | D5×#357, #10599983, #10599984, … |
| maximal quotient of | C2×#65045, C2×#65048, C2×#65049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1426);