Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C32×C5 ⋊ C22⋊C4 |
| Sylow | C32 ⋊ C22⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ #3722 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽16)⋊D3 |
| 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 | |
168 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 37 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,e^a=e^2,g^a=g^3,e^b=e^2,f^b=f^2,g^c=g^442 conjugacy classes — view the full character table.
| maximal subgroup of | #13088350, #13088351, D3×(C2×D3)⋊AGL(1,𝔽5), … |
| maximal normal subgroup of | #13088350, #13088351, D3×(C2×D3)⋊AGL(1,𝔽5), … |
| maximal quotient of | #13085475, #13085476, #13085477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 447);