Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of #10617224⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10617224⋊C3 |
| Fitting | C5×#120 ⋊ C3×C4 |
| derived | C24×C5 ⋊ C2×C3×C4 |
| Hall | #10617224 ⋊ C3 |
| Sylow | C5 ⋊ #2375⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C24×C5 | C2×C3×C4 |
| Frattini | C22 | C2×A4×AGL(1,𝔽5) |
| Fitting | C5×#120 | C3×C4 |
| socle | C22×C5 | C2×C4×A4 |
| prime radical | D5×#120⋊C3 | C2 |
| cosocle | C24×D5 | C22×C3 |
| LCS | G ▷ C24×C5 | |
| UCS | C1 | |
316 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^2,3,2^5,5|a^2=d,e^a=eg,f^a=fgh,i^a=i^2,e^b=egh,f^b=fh,e^c=ef,f^c=e,g^c=h,h^c=gh,i^d=i^444 conjugacy classes — view the full character table.
| maximal subgroup of | #72242815, AGL(1,𝔽5)×#5450, #72243737, … |
| maximal normal subgroup of | AGL(1,𝔽5)×#5450, #72243737, #72253396 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 238231);