Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#4057 |
| Fitting | C2×C5×D8 ⋊ C2 |
| Sylow | C5 ⋊ C2×AGL(1,ℤ/8) |
| 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 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 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)(11,12)(13,14);(1,2)(9,13)(10,14)(12,15);(4,5)(6,7)(9,12)(13,15);(1,2)50 conjugacy classes — view the full character table.
| maximal subgroup of | #10604800, #10604805, C4×#4057, … |
| maximal normal subgroup of | #10604800, #10604805, C4×#4057, … |
| maximal quotient of | C2×#65046, C2×#65074, C2×#65050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1427);