Finite Group DatabasebetaFinite Group Databasebeta| order | 6723 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C83 · Schur multiplier (H2) = C34 |
| direct product | C3×H(3,𝔽3)×C83 |
| unambiguous semidirect product | C32⋊(C32×C83) |
| unambiguous semidirect product | (C32×C83)⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C83 | C32 |
| commutator, Frattini | C3 | C33×C83 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C83 ◁ G | |
112 conjugacy classes of subgroups.
2739 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C83×#6856, C83×#6857, C83×#6859, … |
| maximal normal subgroup of | C83×#6856, C83×#6857, C83×#6859, … |
| maximal quotient of | C3×C32⋊C9×C83, C9×H(3,𝔽3)×C83, C83×C32⋊H(3,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6723, 12);