Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of (C3⋊D3)⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#67154 |
| derived, Hall | C32×C5 ⋊ C22×C4 |
| Sylow | C32 ⋊ C2×C4×D5 |
| Sylow | C5 ⋊ C2×D3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C3⋊D3)⋊D5 |
| commutator, hypocenter | C32×C5 | C22×C4 |
| Frattini | C2 | C2×(C3⋊D3)⋊D5 |
| Fitting, socle | C22×C32×C5 | C22 |
| prime radical | C22×C3×D15 | C2 |
| cosocle | C2×C32×C5 | C23 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
232 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#1200, #13085856, #13085860, … |
| maximal normal subgroup of | Dic3×#1200, #13085856, #13085860, … |
| maximal quotient of | #13087614, #13087615, #13087622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 697);