Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Hall | C32×C5 ⋊ C2×D4 |
| Sylow | C32 ⋊ D4×D5 |
| Sylow | C5 ⋊ D3×D12 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×D5 |
| commutator, Fitting, socle, cosocle | C2×C32×C5 | C23 |
| hypocenter | C32×C5 | C2×D4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
232 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 232 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|a^2=d,b^2=d,c^a=cd,g^a=g^4,c^b=cd,f^b=f^2,e^c=e^257 conjugacy classes — view the full character table.
| maximal subgroup of | #13088915, #13088917, #13088934, … |
| maximal normal subgroup of | #13088915, #13088917, #13088934, … |
| maximal quotient of | #13085677, #13085699, #13085700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 502);