Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D5×C22⋊Q8 |
| Fitting | C5×C22⋊Q8 ⋊ C2 |
| Sylow | C5 ⋊ C2×C22⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×C22⋊Q8 | C2 |
| prime radical | C23×D5 | C22 |
| hypercenter | C22⋊Q8 | D5 |
| hypocenter | C5 | C2×C22⋊Q8 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
322 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 121 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);(10,12,13,17)(11,14,15,16);(6,7)(8,9)(10,11,13,15)(12,16,17,14);(2,3)(4,5)56 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#328, #10599128, #10599129, … |
| maximal normal subgroup of | D5×#328, #10599128, #10599129, … |
| maximal quotient of | #10605597, #10605619, #10605627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1298);