Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | (C2×AGL(1,𝔽5))⋊(C3⋊D3) |
| Hall | C32×C5 ⋊ C22⋊C4 |
| Sylow | C32 ⋊ C22⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ #3752 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D3×AGL(1,𝔽5) |
| commutator, Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×D5×C3⋊D3 | C2 |
| cosocle | C2×C32×D5 | C22 |
| hypocenter | C32×C5 | C22⋊C4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
204 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2^4,3^2,5|a^2=c,b^a=bd,g^a=g^2,e^b=e^2,f^b=f^2,g^c=g^454 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C2×D3)⋊AGL(1,𝔽5), #13088409, #13088414, … |
| maximal normal subgroup of | D3×(C2×D3)⋊AGL(1,𝔽5), #13088409, #13088414, … |
| maximal quotient of | #13085415, #13085416, #13085417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 433);