Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of H(3,𝔽3)×D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C32 |
| direct product | H(3,𝔽3)×SD64 |
| unambiguous semidirect product | (C32×SD64)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×D16 |
| commutator, Frattini, cosocle | C3×C16 | C22×C32 |
| prime radical | H(3,𝔽3)×D16 | C2 |
| LCS | G ▷ C3×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C4×H(3,𝔽3) ◁ C8×H(3,𝔽3) ◁ C16×H(3,𝔽3) ◁ G | |
176 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)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(2,5,6)(4,8,9)(11,25)(12,40)(13,23)(14,38)(15,21)(16,36)(17,19)(18,34)(20,32)(22,30)(24,28)(27,41)(29,39)(31,37)(33,35)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2181);