Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of H(3,𝔽3)×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C32 |
| direct product | H(3,𝔽3)×#368 |
| unambiguous semidirect product | (C32×#368)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C32×D8 |
| commutator | C3×C8 | C2×C32×C4 |
| Frattini, cosocle | C2×C3×C8 | C22×C32 |
| socle | C2×C3 | C32×#107 |
| prime radical | C22×H(3,𝔽3) | Q16 |
| LCS | G ▷ C3×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C4×H(3,𝔽3) ◁ C2×C8×H(3,𝔽3) ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,14,19,16,20,15,30)(12,21,24,36,26,37,25,40)(13,27,28,34,29,41,22,18)(17,31,32,39,33,38,35,23);(2,5,6)(4,8,9)(10,12,22,38,15,25,29,39,16,26,28,31,14,24,13,23)(11,17,27,36,19,32,34,37,20,33,41,40,30,35,18,21)242 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2169);