Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C3⋊(H(7,𝔽2)⋊C5) |
| unambiguous semidirect product | #87574⋊C5 |
| Hall | #87574 ⋊ C5 |
| Sylow | C3 ⋊ H(7,𝔽2)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C24⋊C5 |
| commutator, cosocle, hypocenter | C3×2−1+4 | C22×C5 |
| Fitting | C3×#585 | C2×C5 |
| socle | C2×C3 | C22×C24⋊C5 |
| hypercenter | C4 | D3×C24⋊C5 |
| LCS | G ▷ C3×2−1+4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
1238 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 1238 conjugacy classes of subgroups — too many to draw. Only its 23 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:
2^2,5,2^4,3,2|a^2=i,b^2=i,d^2=i,e^2=i,f^2=i,b^a=bi,h^a=h^2,d^c=de,e^c=df,f^c=defgi,g^c=degi,e^d=ei,f^d=fi,g^d=gi,f^e=fi63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 240282);