Finite Group DatabasebetaFinite Group Databasebeta| order | 7998 |
| automorphism | Aut(G) order 10077480 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D3999 |
| unambiguous semidirect product | C3⋊D1333 |
| unambiguous semidirect product | C31⋊D129 |
| unambiguous semidirect product | C43⋊D93 |
| Fitting, socle, derived, Hall | C3×C31×C43 ⋊ C2 |
| Hall | C31×C43 ⋊ D3 |
| Hall | C3×C43 ⋊ D31 |
| Hall | C3×C31 ⋊ D43 |
| Sylow | C43 ⋊ D93 |
| Sylow | C31 ⋊ D129 |
| Sylow | C3 ⋊ D1333 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C31×C43 | C2 |
| LCS | G ▷ C3×C31×C43 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,31,43|b^a=b^2,c^a=c^30,d^a=d^422001 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D43×D93, D31×D129, D3×D1333, … |
| maximal normal subgroup of | D43×D93, D31×D129, D3×D1333, … |
| maximal quotient of | Dic3999, C2×D3999, D11997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7998, 12);