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) = C1 |
| direct product | 3−1+2×Q64 |
| unambiguous semidirect product | (C9×Q64)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×D16 |
| commutator, Frattini, cosocle | C3×C16 | C22×C32 |
| prime radical | C2×C32 | C3×D16 |
| LCS | G ▷ C3×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C4×3−1+2 ◁ C8×3−1+2 ◁ C16×3−1+2 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 73 points:
(1,2,4,3,6,8,7,5,9)(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)(42,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43);(2,5,6)(4,8,9)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(17,49,33,65)(18,50,34,66)(19,51,35,67)(20,52,36,68)(21,53,37,69)(22,54,38,70)(23,55,39,71)(24,56,40,72)(25,57,41,73)209 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2185);