Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D3)⋊ΓL(1,𝔽16) |
| Hall | C32×C5 ⋊ C22⋊C4 |
| Sylow | C32 ⋊ C22⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ (C2×D3)⋊Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×ΓL(1,𝔽16) |
| commutator, Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C3×D3×D5 | C2 |
| cosocle | C2×C32×D5 | C22 |
| hypocenter | C32×C5 | C22⋊C4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
152 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,b^a=bd,f^a=f^2,g^a=g^3,e^b=e^2,g^c=g^448 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C2×D3)⋊AGL(1,𝔽5), #13088405, #13088408, … |
| maximal normal subgroup of | D3×(C2×D3)⋊AGL(1,𝔽5), #13088405, #13088408, … |
| maximal quotient of | #13085446, #13085447, #13085448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 440);