Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived | C5×SL(2,𝔽3) ⋊ C42 |
| Sylow | C5 ⋊ #73063 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | AGL(1,𝔽5)×S4 |
| commutator, hypocenter | C5×SL(2,𝔽3) | C42 |
| Fitting | C2×C5×Q8 | C4×D3 |
| socle | C22×C5 | C4×S4 |
| prime radical, cosocle | C2×D5×SL(2,𝔽3) | C22 |
| LCS | G ▷ C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
354 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^4,3,5,2^3|a^2=c,b^2=d,g^2=i,h^2=i,b^a=bi,f^a=f^2,e^b=e^2,g^b=gh,h^b=hi,f^c=f^4,g^e=h,h^e=ghi,h^g=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 238089);