Finite Group DatabasebetaFinite Group Databasebeta| order | 6723 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C83 · Schur multiplier (H2) = C3 |
| direct product | #731×C83 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C83 | H(3,𝔽3) |
| commutator, Frattini | C32 | C32×C83 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C83 ◁ C32×C83 ◁ G | |
28 conjugacy classes of subgroups.
1411 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×#731×C83, C83×#6908, C83×#6910, … |
| maximal normal subgroup of | C3×#731×C83, C83×#6908, C83×#6910, … |
| maximal quotient of | C83×#6857, C83×#6858, C83×#6859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6723, 8);