Finite Group DatabasebetaFinite Group Databasebeta| order | 6996 |
| automorphism | Aut(G) order 3637920 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic1749 |
| unambiguous semidirect product | C3⋊Dic583 |
| unambiguous semidirect product | C11⋊Dic159 |
| derived, Hall | C3×C11×C53 ⋊ C4 |
| Hall | C11×C53 ⋊ Dic3 |
| Hall | C3×C53 ⋊ Dic11 |
| Sylow | C53 ⋊ Dic33 |
| Hall | C3×C11 ⋊ Dic53 |
| Sylow | C11 ⋊ Dic159 |
| Sylow | C3 ⋊ Dic583 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D1749 |
| commutator, hypocenter | C3×C11×C53 | C4 |
| Fitting, socle | C2×C3×C11×C53 | C2 |
| LCS | G ▷ C3×C11×C53 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,11,53,2|a^2=e,b^a=b^2,c^a=c^10,d^a=d^521752 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic5247, C3⋊Dic1749, C3×Dic1749, … |
| maximal normal subgroup of | C3×Dic1749, C5×Dic1749, C3⋊((C11⋊C5)⋊Dic53), … |
| maximal quotient of | Dic5247, C3⋊Dic1749, C3×Dic1749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6996, 8);