Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C32×C5 ⋊ #41 |
| Sylow | C32 ⋊ #709 |
| Sylow | C5 ⋊ #3798 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×D15 |
| commutator, socle, cosocle | C2×C32×C5 | C23 |
| Frattini | C2 | C2×D3×D15 |
| Fitting | C32×C4×C5 | C22 |
| hypocenter | C32×C5 | #41 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C4 | |
176 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^4,3^2,5|c^2=d,b^a=bd,f^a=f^2,g^a=g^4,e^b=e^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #13085660, #13085661, #13085662, … |
| maximal normal subgroup of | #13085660, #13085661, #13085662, … |
| maximal quotient of | #13087312, #13087317, #13087320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 668);