Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | D5×#124 |
| Fitting | C5×#124 ⋊ C2 |
| Sylow | C5 ⋊ C2×#124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#124 | C2 |
| prime radical | C2×D4×D5 | C2 |
| hypercenter | #124 | D5 |
| hypocenter | C5 | C2×#124 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #124 | |
330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 330 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)(6,8,9,7)(11,15)(14,17);(6,7,9,8)(10,12,13,16)(11,14,15,17);(7,8)(10,11)(12,14)(13,15)(16,17);(2,3)(4,5)56 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#329, #10599136, #10599140, … |
| maximal normal subgroup of | D5×#329, #10599136, #10599140, … |
| maximal quotient of | #10605880, #10605895, #10605898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1345);