Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D5×#135 |
| Fitting | C5×#135 ⋊ C2 |
| Sylow | C5 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D5 |
| commutator, cosocle | C4×C5 | C24 |
| Frattini | C4 | C23×D5 |
| Fitting | C5×#135 | C2 |
| socle | C2×C5 | C22×D4 |
| hypercenter | #135 | D5 |
| hypocenter | C5 | C2×#135 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,18,20,21,19);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16);(18,19)(20,21)56 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#359, #10600034, D5×#360, … |
| maximal normal subgroup of | D5×#359, #10600034, D5×#360, … |
| maximal quotient of | #10610549, #10610553, #10610556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1439);