Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #625 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×Dic3⋊Dic3 |
| unambiguous semidirect product | Dic3⋊(C5×Dic3) |
| Hall | C32×C5 ⋊ C4⋊C4 |
| Sylow | C32 ⋊ C5×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | D32 |
| commutator | C2×C32 | C2×C4×C5 |
| Frattini | C22 | C5×D32 |
| Fitting, socle, prime radical | C22×C32×C5 | C22 |
| cosocle | C22×C32 | C22×C5 |
| hypocenter | C32 | C5×C4⋊C4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C5 | |
120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 240 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4,5)(6,7,8,9)(11,13)(15,16)(17,18,19);(10,11,12,13)(14,15,16)(18,19)150 conjugacy classes — view the full character table.
| maximal subgroup of | D5×Dic3⋊Dic3, #13085869, #13085874, … |
| maximal normal subgroup of | D5×Dic3⋊Dic3, #13085869, #13085874, … |
| maximal quotient of | C5×Dic3⋊(C3⋊C8), C5×#63573, C5×#63575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 259);