Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of #857⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #857⋊AGL(1,𝔽5) |
| Fitting | C42×C5 ⋊ C4×D3 |
| derived | C5×C42⋊C3 ⋊ C2×C4 |
| Sylow | C5 ⋊ #857⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C5×C42⋊C3 | C2×C4 |
| Frattini | C22 | AGL(1,𝔽5)×S4 |
| Fitting | C42×C5 | C4×D3 |
| socle | C22×C5 | C4×S4 |
| prime radical | D5×#857 | C2 |
| cosocle | D5×C42⋊C3 | C22 |
| LCS | G ▷ C5×C42⋊C3 | |
| UCS | C1 | |
294 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 60 |
Polycyclic presentation:
2^3,3,5,2^4|a^2=c,f^2=h,g^2=i,e^a=e^2,f^a=fh,g^a=gi,d^b=d^2,f^b=fgi,g^b=gi,h^b=hi,e^c=e^4,f^d=gh,g^d=fg,h^d=i,i^d=hi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72253403, AGL(1,𝔽5)×#857⋊C2 |
| maximal normal subgroup of | AGL(1,𝔽5)×#857⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 238046);