Finite Group DatabasebetaFinite Group Databasebeta| order | 1715 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C72 · Schur multiplier (H2) = C1 |
| direct product | C5×7−1+2 |
| unambiguous semidirect product | C49⋊(C5×C7) |
| unambiguous semidirect product | (C5×C49)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C7 | C72 |
| commutator, Frattini, cosocle | C7 | C5×C72 |
| prime radical | C5×C72 | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C5×C7 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 54 |
| transitive degree | μt(G) = 245 |
A minimal faithful permutation representation — 2 generators on 54 points:
(1,2,3,4,5)(6,7,9,13,20,31,40,8,11,15,22,33,42,49,12,18,25,35,44,41,50,19,29,24,37,32,43,51,30,28,27,21,34,45,52,39,17,14,23,36,46,53,48,10,16,26,38,47,54);(7,10,17,28,29,18,11)(9,14,24,15,16,27,25)(13,21,22,23,35,26,37)(20,32,38,44,36,33,34)(31,41,45,47,42,43,46)(40,49,50,51,52,53,54)275 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1715, 4);