Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C5⋊GL(2,𝔽3)) |
| unambiguous semidirect product | C5⋊(C3⋊GL(2,𝔽3)) |
| Fitting | C3×C5×Q8 ⋊ D3 |
| derived | C3×C5×SL(2,𝔽3) ⋊ C2 |
| Sylow | C5 ⋊ C3⋊GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊(C5⋊S4) |
| commutator, cosocle, hypocenter | C3×C5×SL(2,𝔽3) | C2 |
| Fitting | C3×C5×Q8 | D3 |
| socle | C2×C3×C5 | S4 |
| LCS | G ▷ C3×C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2,3^2,5,2^3|e^2=g,f^2=g,b^a=b^2,c^a=c^2,d^a=d^4,e^a=fg,f^a=eg,e^b=ef,f^b=e,f^e=fg57 conjugacy classes — view the full character table.
| maximal subgroup of | #13088533, #13088534, #13088537, … |
| maximal normal subgroup of | #13088533, #13088534, #13088537, … |
| maximal quotient of | C3⋊#91184, C2×C3⋊(C5⋊GL(2,𝔽3)) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 536);