Finite Group DatabasebetaFinite Group Databasebeta| order | 24000 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C4×C16×C25 (order 6400) · Inn(G) = C1 · Out(G) = C22×C4×C16×C25 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C24000 |
| abelian | C3×C64×C125 |
| multiplicative | (ℤ/24001)ˣ, (ℤ/48002)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C25×C32 | C2×C3×C5 |
| socle, prime radical | C2×C3×C5 | C25×C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
56 conjugacy classes of subgroups.
24000 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | Dic12000, D24000 |
| maximal subgroup of | Dic12000, D24000 |
| maximal normal subgroup of | Dic12000, D24000 |
| automorphism group of | C24001 |
| outer automorphism group of | C24001 |
Past the SmallGroups library coverage — no SmallGroups id.