Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #10617747⋊C3 |
| Hall | #10617747 ⋊ C3 |
| Sylow | C5 ⋊ C4⋊#994 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5×A4 |
| commutator, cosocle | C2×C5×Q8 | C23×C3 |
| Fitting | C4×C5×#41 | C2×C3 |
| socle | C22×C5 | C23×A4 |
| hypercenter | C42 | D5×A4 |
| hypocenter | C5×Q8 | C2×C3×D4 |
| LCS | G ▷ C2×C5×Q8 ▷ C5×Q8 | |
| UCS | C1 ◁ C22 ◁ C42 | |
786 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 786 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 480 |
Polycyclic presentation:
2^3,3,2^3,5,2|a^2=i,b^2=i,c^2=e,f^2=i,g^2=i,b^a=bi,c^a=ce,h^a=h^4,f^d=fg,g^d=f,g^f=gi164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 239724);