Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D5×#123 |
| Fitting | C5×#123 ⋊ C2 |
| Sylow | C5 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#123 | C2 |
| prime radical | C2×D4×D5 | C2 |
| hypercenter | #123 | D5 |
| hypocenter | C5 | C2×#123 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #123 | |
342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 342 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) = 17 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,4,5,3)(7,9)(10,12,13,16)(11,15,14,17);(11,14)(15,17);(6,7)(8,9)(10,11)(12,15)(13,14)(16,17);(2,3)(4,5)56 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#352, D5⋊#352, #10604533, … |
| maximal normal subgroup of | D5×#352, D5⋊#352, #10604533, … |
| maximal quotient of | #10605783, #10605797, #10605799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1324);