Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C3⋊ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊(C3⋊#170) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊#170) |
| derived, Hall | C32×C5 ⋊ C2×C8 |
| Sylow | C32 ⋊ C2×#170 |
| Sylow | C5 ⋊ C2×C3⋊(C3⋊C8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊ΓL(1,𝔽16) |
| commutator, hypocenter | C32×C5 | C2×C8 |
| Frattini | C2 | C2×C3⋊ΓL(1,𝔽16) |
| Fitting, socle, prime radical | C22×C32×C5 | C4 |
| cosocle | C32×Dic5 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
132 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊Dic3×#170, #13085433, #13085434, … |
| maximal normal subgroup of | C3⋊Dic3×#170, #13085433, #13085434, … |
| maximal quotient of | C2×C3⋊(C3⋊#674), #13086300, C4×C3⋊(C3⋊#170), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 565);