Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D4×Dic10 |
| Sylow | C5 ⋊ D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C4×C5×D4 | C2 |
| prime radical | C2×C5×D4 | C22 |
| hypercenter | C4×D4 | D5 |
| hypocenter | C5 | D4×Q8 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 160 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,4,5,3)(10,12,13,17)(11,14,15,16);(2,3)(4,5)(10,11,13,15)(12,16,17,14);(7,9);(6,7)(8,9)65 conjugacy classes — view the full character table.
| maximal subgroup of | #10598980, #10598982, #10598985, … |
| maximal normal subgroup of | #10598980, #10598982, #10598985, … |
| maximal quotient of | C22⋊C4×Dic10, #10605196, #10605197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1209);