Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2 |
| direct product | 3−1+2×#479 |
| unambiguous semidirect product | (C9×#479)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C32×D4 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C22×C32 | C2×C3×D4 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C42×3−1+2 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,4,3,6,8,7,5,9)(10,11,13,15)(12,16,17,14)(18,19,21,23)(20,26,27,33)(22,30,29,25)(24,31,32,28);(2,5,6)(4,8,9)(10,12,13,17)(11,14,15,16)(19,23)(24,32)(25,30)(26,33);(18,20,21,27)(19,24,23,32)(22,31,29,28)(25,33,30,26)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11260);