Finite Group DatabasebetaFinite Group Databasebeta| order | 1701 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C7×C27 · Schur multiplier (H2) = C1 |
| direct product | C7×C81⋊C3 |
| unambiguous semidirect product | C81⋊(C3×C7) |
| subgroup | G/H | |
|---|---|---|
| center | C7×C27 | C32 |
| commutator | C3 | C3×C7×C27 |
| Frattini, cosocle | C27 | C32×C7 |
| socle | C3×C7 | C3×C27 |
| prime radical | C32×C7 | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C7×C27 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 88 |
| transitive degree | μt(G) = 567 |
A minimal faithful permutation representation — 2 generators on 88 points:
(1,2,3,4,5,6,7)(8,9,13,10,15,25,18,30,44,11,16,26,19,31,45,35,52,66,21,33,46,37,54,67,57,73,81,12,17,24,20,32,42,36,53,63,22,34,43,38,55,64,58,74,79,40,56,65,60,75,80,76,85,87,23,14,27,39,28,47,59,49,68,41,29,48,61,50,69,77,71,82,62,51,70,78,72,83,86,84,88);(9,14,17)(13,24,27)(15,28,32)(16,29,34)(25,42,47)(26,43,48)(30,49,53)(31,50,55)(33,51,56)(44,63,68)(45,64,69)(46,65,70)(52,71,74)(54,72,75)(66,79,82)(67,80,83)(73,84,85)(81,87,88)693 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1701, 24);