Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C2×C5×Q8⋊C9 |
| unambiguous semidirect product | (C2×Q8)⋊(C5×C9) |
| unambiguous semidirect product | (C2×C5×Q8)⋊C9 |
| derived | Q8 ⋊ C2×C5×C9 |
| Hall | C2×C5×Q8 ⋊ C9 |
| Sylow | C2×Q8 ⋊ C5×C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical, hypercenter | C22×C3×C5 | A4 |
| commutator, hypocenter | Q8 | C2×C5×C9 |
| Frattini | C2×C3 | C2×C5×A4 |
| Fitting | C2×C3×C5×Q8 | C3 |
| cosocle | C3×Q8 | C2×C3×C5 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C3×C5 | |
58 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
210 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#63421, #13083943, Dic5×Q8⋊C9, … |
| maximal normal subgroup of | C5×#63421, #13083943, Dic5×Q8⋊C9, … |
| maximal quotient of | C4×C5×Q8⋊C9, C22×C5×Q8⋊C9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 153);