Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10605017⋊C3 |
| Fitting, derived | C5×H(5,𝔽2) ⋊ C3×C4 |
| Hall | #10605017 ⋊ C3 |
| Sylow | C5 ⋊ #1790⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #65800⋊C3 |
| commutator, Fitting, hypocenter | C5×H(5,𝔽2) | C3×C4 |
| Frattini | C23 | A4×AGL(1,𝔽5) |
| socle | C2×C5 | #379⋊C3 |
| prime radical | D5×#997 | C2 |
| cosocle | D5×H(5,𝔽2) | C2×C3 |
| LCS | G ▷ C5×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
310 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 40 |
Polycyclic presentation:
2,3,2^5,5,2|a^2=c,d^a=df,e^a=efg,h^a=h^2,d^b=de,e^b=d,f^b=gi,g^b=fgi,h^c=h^4,g^d=gi,f^e=fi,g^e=gi43 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×B(4,𝔽2)⋊C3, #72244488, #72244492, … |
| maximal normal subgroup of | AGL(1,𝔽5)×B(4,𝔽2)⋊C3, #72244488, #72244492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237304);