Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#4081 |
| Sylow | C5 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D5 |
| commutator, cosocle | C22×C5 | C24 |
| Frattini | C22 | C23×D5 |
| Fitting, prime radical | C22×C5×D4 | C2 |
| socle | C23×C5 | C23 |
| hypercenter | C22×D4 | D5 |
| hypocenter | C5 | C2×#123 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 19 points:
(3,4)(5,6)(7,8,10,11,9)(12,14)(13,16)(15,18)(17,19);(1,2)(14,18)(16,19);(4,6)(8,9)(10,11)(12,13,15,17)(14,16,18,19);(1,2)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#64692, #10600816, #10600822, … |
| maximal normal subgroup of | C2×#64692, #10600816, #10600822, … |
| maximal quotient of | C2×#65232, C2×#65235, C2×#65233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1468);