Finite Group DatabasebetaFinite Group Databasebeta| order | 24000 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 1024000 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C4×C5 |
| abelian | C3×C4×C5×C16×C25 |
| multiplicative | (ℤ/24341)ˣ, (ℤ/24461)ˣ, (ℤ/30025)ˣ, (ℤ/30125)ˣ, (ℤ/48682)ˣ, (ℤ/48922)ˣ, (ℤ/60050)ˣ, (ℤ/60250)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C5×C8 | C22×C3×C52 |
| socle, prime radical | C22×C3×C52 | C2×C5×C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
812 conjugacy classes of subgroups.
24000 conjugacy classes. The character table is too large to display.
| automorphism group of | C101×C241, C61×C401 |
| outer automorphism group of | C101×C241, C61×C401 |
Past the SmallGroups library coverage — no SmallGroups id.