Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of H(3,𝔽3)×#471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22 |
| direct product | 3−1+2×#473 |
| unambiguous semidirect product | (C9×#473)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#121 |
| commutator, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| prime radical | C32×#41 | C22×C3 |
| LCS | G ▷ C2×C3×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C4×3−1+2 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,4,3,6,8,7,5,9)(10,11,16,21)(12,22,26,32)(13,27,29,35)(14,20,15,19)(17,25,33,24)(18,30,34,28)(23,37,39,31)(36,40,41,38);(2,5,6)(4,8,9)(10,12,15,25,16,26,14,24)(11,17,20,32,21,33,19,22)(13,23,30,40,29,39,28,38)(18,31,35,41,34,37,27,36);(10,13)(11,18)(12,23)(14,28)(15,30)(16,29)(17,31)(19,27)(20,35)(21,34)(22,36)(24,38)(25,40)(26,39)(32,41)(33,37)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11242);