Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C5⋊((C22⋊C9)⋊C2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×C5⋊((C22⋊C9)⋊C2) |
| unambiguous semidirect product | (C2×C5)⋊((C22⋊C9)⋊C2) |
| unambiguous semidirect product | (C2×C22⋊C9)⋊D5 |
| derived | C5×C22⋊C9 ⋊ C22 |
| Sylow | C5 ⋊ C2×(C22⋊C9)⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C5⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C5×C22⋊C9 | C22 |
| Frattini | C3 | C2×C5⋊S4 |
| Fitting, socle | C23×C3×C5 | D3 |
| LCS | G ▷ C5×C22⋊C9 | |
| UCS | C1 ◁ C2 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
66 conjugacy classes — view the full character table.
| maximal subgroup of | #13085156, #13085157, #13085160, … |
| maximal normal subgroup of | #13085156, #13085157, #13085160, … |
| maximal quotient of | #13085203, C4×C5⋊((C22⋊C9)⋊C2), C4⋊(C5⋊((C22⋊C9)⋊C2)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 397);