Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#3804 |
| Hall | C32×C5 ⋊ #41 |
| Sylow | C32 ⋊ C5×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×D32 |
| commutator, cosocle | C2×C32 | C23×C5 |
| Frattini | C2 | C2×C5×D32 |
| Fitting | C22×C32×C5 | C22 |
| socle | C2×C32×C5 | C23 |
| hypercenter | C22×C5 | D32 |
| hypocenter | C32 | C5×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
176 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 120 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3,4,5)(7,11)(10,13)(14,15,16)(17,18,19);(6,8,9,12)(7,10,11,13)(15,16);(6,7)(8,10)(9,11)(12,13)(18,19)135 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3804, #13088975, #13088979, … |
| maximal normal subgroup of | D5×#3804, #13088975, #13088979, … |
| maximal quotient of | C5×#63846, C5×#63847, C5×#63848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 653);