Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Hall | C32×C5 ⋊ #41 |
| Sylow | C32 ⋊ #709 |
| Sylow | C5 ⋊ #3830 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×C3⋊D3 |
| commutator, socle, cosocle | C2×C32×C5 | C23 |
| Fitting | C22×C32×C5 | C22 |
| hypercenter | C22 | D5×C3⋊D3 |
| hypocenter | C32×C5 | #41 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^4,3^2,5|b^2=d,c^a=cd,g^a=g^4,e^b=e^2,f^b=f^278 conjugacy classes — view the full character table.
| maximal subgroup of | #13088794, #13088795, D3×#6779, … |
| maximal normal subgroup of | #13088794, #13088795, D3×#6779, … |
| maximal quotient of | #13086769, #13086770, #13086771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 631);