Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(C5⋊(#41⋊C3)) |
| unambiguous semidirect product | C5⋊(C3⋊(#41⋊C3)) |
| unambiguous semidirect product | #6821⋊C3 |
| Sylow | C5 ⋊ C3⋊(#41⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×D15 |
| commutator, Fitting, cosocle, hypocenter | C3×C5×Q8 | C2×C3 |
| socle | C2×C3×C5 | C2×A4 |
| LCS | G ▷ C3×C5×Q8 | |
| UCS | C1 ◁ C2 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2,3,2^2,3,5,2|a^2=g,c^2=g,d^2=g,e^a=e^2,f^a=f^4,c^b=cd,d^b=c,d^c=dg63 conjugacy classes — view the full character table.
| maximal subgroup of | #13088546, #13088547, #13088548, … |
| maximal normal subgroup of | #13088546, #13088547, #13088548, … |
| maximal quotient of | SL(2,𝔽3)×Dic15, C2×C3⋊(C5⋊(#41⋊C3)) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 544);