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 ⋊ #685 |
| Sylow | C5 ⋊ (C2×D3)⋊Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | (C3⋊D3)⋊D5 |
| commutator | C2×C32×C5 | C2×C4 |
| Fitting, socle, cosocle | C22×C32×C5 | C22 |
| prime radical | C22×C3×D15 | C2 |
| hypocenter | C32×C5 | C22⋊C4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2^4,3^2,5|b^2=c,b^a=bd,e^a=e^2,g^a=g^4,e^b=e^2,f^b=f^278 conjugacy classes — view the full character table.
| maximal subgroup of | D5×(C2×D3)⋊Dic3, D3×#6665, D3×#6667, … |
| maximal normal subgroup of | D5×(C2×D3)⋊Dic3, D3×#6665, D3×#6667, … |
| maximal quotient of | #13084889, #13084891, #13084893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 298);