Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#3811 |
| Fitting | C22×C32×C5 ⋊ C22 |
| Hall | C32×C5 ⋊ C2×D4 |
| Sylow | C32 ⋊ C2×C5×D4 |
| 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 | C2×C5×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
248 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,12)(13,14,15);(7,9)(11,12);(7,9)(14,15)120 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#63738, #13085592, C5×#64233, … |
| maximal normal subgroup of | C5×#63738, #13085592, C5×#64233, … |
| maximal quotient of | C5×#63883, C5×#63910, C5×#63896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 660);