Finite Group DatabasebetaFinite Group Databasebeta| order | 31720 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C32×C43×C5 (order 11520) · Inn(G) = C1 · Out(G) = C22×C32×C43×C5 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C31720 |
| abelian | C5×C8×C13×C61 |
| multiplicative | (ℤ/31721)ˣ, (ℤ/63442)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4 | C2×C5×C13×C61 |
| socle, prime radical | C2×C5×C13×C61 | C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
32 conjugacy classes of subgroups.
31720 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | Dic15860, D31720 |
| maximal subgroup of | Dic15860, D31720 |
| maximal normal subgroup of | Dic15860, D31720 |
| automorphism group of | C31721 |
| outer automorphism group of | C31721 |
Past the SmallGroups library coverage — no SmallGroups id.