Finite Group DatabasebetaFinite Group Databasebeta| order | 6972 |
| automorphism | Aut(G) order 3430224 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D3486 |
| direct product | C2×D1743 |
| unambiguous semidirect product | (C2×C3)⋊D581 |
| unambiguous semidirect product | (C2×C7)⋊D249 |
| unambiguous semidirect product | (C2×C83)⋊D21 |
| Fitting, socle | C2×C3×C7×C83 ⋊ C2 |
| derived, Hall | C3×C7×C83 ⋊ C22 |
| Hall | C7×C83 ⋊ C2×D3 |
| Hall | C3×C83 ⋊ C2×D7 |
| Sylow | C83 ⋊ C2×D21 |
| Hall | C3×C7 ⋊ C2×D83 |
| Sylow | C7 ⋊ C2×D249 |
| Sylow | C3 ⋊ C2×D581 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D1743 |
| commutator, hypocenter | C3×C7×C83 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7×C83 | C2 |
| LCS | G ▷ C3×C7×C83 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
1746 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×D5229, C2×C3⋊(ΓL(1,𝔽8)⋊D83), D3×D1743, … |
| maximal normal subgroup of | C2×C3⋊(ΓL(1,𝔽8)⋊D83), C2×C3×D1743, C2×C5×D1743, … |
| maximal quotient of | C2×D5229, D3×D1743, C2×C3×D1743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6972, 28);