Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C5×C22⋊SL(2,𝔽3) ⋊ C4 |
| Sylow | C5 ⋊ #68001 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11746597 |
| commutator, hypocenter | C5×C22⋊SL(2,𝔽3) | C4 |
| Frattini | C24 | C5⋊S4 |
| Fitting | C5×#393 | D3 |
| socle | C23×C5 | SL(2,ℤ/4) |
| prime radical, cosocle | C5×#393⋊C3 | C2 |
| LCS | G ▷ C5×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
144 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 144 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) = 120 |
Polycyclic presentation:
2^2,3,5,2^5|a^2=b,e^2=i,f^2=i,c^a=c^2,d^a=d^4,e^a=efg,f^a=fghi,g^a=gh,e^b=eg,f^b=fh,e^c=f,f^c=efi,g^c=h,h^c=gh,f^e=fi57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237233);