Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #73002 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1020⋊D5 |
| commutator, cosocle, hypocenter | C5×C22⋊SL(2,𝔽3) | C22 |
| Frattini | C23 | D5×S4 |
| Fitting | C22×C5×Q8 | C2×D3 |
| socle | C23×C5 | C2×S4 |
| prime radical | C5×#5931 | C2 |
| LCS | G ▷ C5×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
240 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 240 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) = 240 |
Polycyclic presentation:
2^2,3,5,2^5|a^2=i,e^2=i,f^2=i,d^a=d^4,e^a=eg,f^a=fh,c^b=c^2,e^b=ef,f^b=fi,g^b=gh,e^c=f,f^c=efi,g^c=h,h^c=gh,f^e=fi51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237513);