Finite Group DatabasebetaFinite Group Databasebeta| order | 12500 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C4×C625 (order 5000) · Inn(G) = C1 · Out(G) = C2×C4×C625 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C12500 |
| abelian | C4×C3125 |
| multiplicative | (ℤ/15625)ˣ, (ℤ/31250)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C625 | C2×C5 |
| socle, prime radical | C2×C5 | C2×C625 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
18 conjugacy classes of subgroups.
12500 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id.