Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23 |
| direct product | 3−1+2×#389 |
| unambiguous semidirect product | (C9×#389)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23×C3 | C23×C32 |
| commutator | C22×C3 | C22×C32×C4 |
| prime radical | C23×C32 | C23×C3 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,3,6,8,7,5,9)(11,13)(15,17)(18,19,21,20);(2,5,6)(4,8,9)(14,15)(16,17)(18,20,21,19);(10,11,12,13)308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10990);